1

THE ABSOLUTE RELATIONS
OF TIME AND SPACE

BY
A. A. ROBB

THE LIBRARY

OF

THE UNIVERSITY
OF CALIFORNIA

LOS ANGELES

THE ABSOLUTE RELATIONS
OF TIME AND SPACE

CAMBRIDGE UNIVERSITY PRESS

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THE ABSOLUTE RELATIONS
OF TIME AND SPACE

BY

ALFRED A. ROBB, Sc.D., D.Sc., PH.D., F.R.S.

CAMBRIDGE

AT THE UNIVERSITY PRESS
1921

First Edition, 1921
Reprinted, 1921

PKEFACE

AT the meeting of the British Association in 1902, Lord Rayleigh
gave a paper entitled "Does motion through the ether cause double
refraction?" in which he described some experiments which seemed
to indicate that the answer was in the negative. I recollect that
on this occasion Professor Larmor was asked whether he would
expect any such effect and he replied that he did not expect any.

In the discussion which followed reference was made to the
null results of all attempts to detect uniform motion through the
ether and to the way in which things seemed to conspire together
to give these null results.

The impression made on me by this discussion was: that in
order properly to understand what happened, it would be necessary
to be quite clear as to what we mean by equality of lengths, etc.,
and I decided that I should try at some future time to carry out
an analysis of this subject.

I am not certain that I had not some idea of doing this even
before the British Association meeting, but in any case, the inspira-
tion came from Sir Joseph Larmor, either at this meeting or on
some previous occasion while attending his lectures.

Some years later I proceeded to try to carry out this idea, and
while engaged in endeavouring to solve the problem, I heard for
the first time of Einstein's work.

From the first I felt that Einstein's standpoint and method of
treatment were unsatisfactory, though his mathematical transfor-
mations might be sound enough, and I decided to proceed in my
own way in search of a suitable basis for a theory.

In particular I felt strongly repelled by the idea that events
could be simultaneous to one person and not simultaneous to
another; which was one of Einstein's chief contentions.

This seemed to destroy all sense of the reality of the external
world and to leave the physical universe no better than a dream,
or rather, a nightmare.

766548

vi PREFACE

If two physicists A and B agree to discuss a physical experi-
ment, their agreement implies that they admit, in some sense, a
common world in which the experiment is supposed to take place.

It might be urged perhaps that we have merely got a corre-
spondence between the physical worlds of A and B, but if so, where,
or how, does this correspondence subsist?

It cannot be in A's mind alone, or it would not be a corre-
spondence, and similarly it cannot be in B's mind alone.

It seems to follow that it must be in some common sub-stratum;
and this brings us at once back to an objective standpoint.

The first work which I published on this subject was a short
tract entitled Optical Geometry of Motion, a New View of the
Theory of Relativity, which appeared in 1911.

This paper, though it did not claim to give a complete logical
analysis of the subject, yet contained some of the germs of my
later work and, in particular, it avoided any attempt to identify
instants at different places. Later on the idea of "Conical Order"
occurred to me, in which such instants are treated as definitely
distinct.

The working out of this idea was a somewhat lengthy task and
in 1913 I published a short preliminary account of it under the
title A Theory of Time and Space, which was also the title of a
book on this subject on which I was then engaged.

This book was in the press at the time of the outbreak of the
war and was finally published towards the end of 1914.

Unhappily at that period people were concerning themselves
rather with trying to sever one another's connexions with Time
and Space altogether, than with any attempt to understand such
things; so that it was hardly an ideal occasion to bring out a book
on the subject.

The subject moreover was not an easy one, and I have been
told more than once that my book is difficult reading.

To this I can only reply as did Mr Oliver Heaviside, under similar
circumstances, that it was perhaps even more difficult to write.

Be that as it may, the results arrived at fully justified my
attitude towards Einstein's standpoint.

PREFACE vii

I succeeded in developing a theory of Time and Space in terms
of the relations of before and after, but in which these relations are
regarded as absolute and not dependent on the particular observer.

In fact it is not a "theory of relativity" at all in Einstein's
sense, although it certainly does involve relations.

These relations of before and after, serving, as they do, as a
physical basis for the mathematical theory, were quite ignored in
Einstein's treatment; with the result that the absolute features
were lost sight of.

Even now, some six years from the date of publication of my
book, comparatively few of Einstein's followers appear to realize
the extreme importance of these relations, or to recognize how
they alter the entire aspect of the subject.

The theory, in so far as its postulates have an interpretation,
becomes a physical theory in the ordinary sense, but these postulates
are used to build up a pure mathematical structure.

From the physical standpoint the question is: whether the
postulates as interpreted are correct expressions of physical facts,
or in some respect only approximations?

If the postulates are not all correct expressions of the facts,
then which of them require emendation and what emendation do
they require?

As regards the pure mathematical aspect of the theory: this
of course remains unaffected by the physical interpretation of the
postulates, and those who are interested only in pure mathematics
may find that the method employed has certain advantages as
a study of the foundations of geometry.

In particular it may be noticed that by this method we get
a system of geometry in which "congruence" appears, not as
something extraneous grafted on to an otherwise complete system,
but as an intrinsic part of the system itself.

I had intended making further developments of this theory,
but the outbreak of the war caused an interruption of my work.

In the meantime Einstein produced his "generalized relativity"
theory and the reader will doubtless wish to know how this work
bears upon it.

viii PREFACE

So far as I can at present judge, the situation is this: once
coordinates have been introduced, the theory here developed gives
rise to the same analysis as Einstein's so-called "restricted re-
lativity" and this latter cannot be regarded as satisfactory apart
from my work, or some equivalent.

Einstein's more recent work is extremely analytical in character.

The before and after relations have not been employed at all in
its foundation, although it is evident that, if these relations are
a sufficient basis for the simple theory, they must play an equally
important part in any generalization. Moreover these relations
most certainly have a physical significance whatever theory be the
correct one.

A generalization of my own work is evidently possible and, to
a certain extent, I can see a method of carrying this out, although
I have not as yet worked out the details. (See Appendix.)

In the meantime it seemed desirable to write some sort of
introduction to my Theory of Time and Space which, while not
going into the proofs of theorems, would yet convey to a larger
circle of readers the main results arrived at in that work.

A. A. R.

CAMBRIDGE,

November 12, 1920.

CONTENTS

PAGE

PRELIMINARY CONSIDERATIONS ... 1

CONICAL ORDER 16

NORMALITY OF GENERAL LINES HAVING A

COMMON ELEMENT 46

THEORY OF CONGRUENCE . . . .56

INTRODUCTION OF COORDINATES ... 72

INTERPRETATION OF RESULTS ... 76

APPENDIX 78

THE ABSOLUTE RELATIONS OF TIME
AND SPACE

PRELIMINARY CONSIDERATIONS

THE study of Time and Space is one which in certain respects is
extremely elusive and involves a number of difficulties which in
ordinary daily life we are apt to overlook.

In scientific work, however, it is all-important to have clear
ideas and to know exactly what our statements mean.

This is by no means always an easy task, for it frequently
happens that our crude ideas on certain things may be sufficiently
precise for certain purposes, but not precise enough for others.

Thus in the ordinary elementary teaching of plane geometry
there are certain difficulties which are generally passed over,
largely because they are real difficulties and a proper understanding
of them could hardly be expected from a beginner.

For instance the use of ruler and compasses and the method of
superposition.

The use of the ruler conveys a somewhat crude idea of what
we mean in the physical world by points lying in a straight line,
while the use of compasses conveys an equally crude idea of what
we mean by points in a plane being equally distant from a given
point in the plane.

The method of superposition involves ideas which are closely
akin to those involved in the supposed use of compasses, but of
a more elaborate character.

Both sets of ideas may be described as ideas of congruence.

Although there are other difficulties besides these to be over-
come, still these will suffice for our present purpose, which is to
show that certain points have been slurred over when we first
began the study of geometry, which later on may require further
elucidation.

Now let us approach this subject as a beginner of sufficient
intelligence might be supposed to do.

There is one thing which we might observe, namely : that though
we make use of figures drawn on paper to assist us in keeping the

K. 1

2 TIME AND SPACE

facts in mind, yet in proving a theorem, as distinguished from
making use of the result, there is no necessity that the figure
should be accurately drawn. A very rough figure will suffice and,
if we are fairly expert, and the theorem not too complicated, we
can dispense with a figure altogether.

Next let us suppose the figures to be accurately drawn on a
plane sheet of paper (whatever the expressions "accurately drawn"
and "plane" may mean) and then suppose the sheet of paper to
be rolled up into a spiral, we could still make use of the figures
on the curved sheet as mental images in proving our theorems,
although our original straight lines would now (with certain
exceptions) be no longer straight.

We could however substitute for our ruler a flexible string,
drawn taut, so as to lie in contact with the curved surface of the
paper and similarly we could make use of a flexible inextensible
tape line or string instead of our original compasses and all our
theorems would work out as before, except that lines would be
curved which had originally been straight and lengths would be
measured along such lines instead of "directly" between points.

With such modifications, to every theorem concerning figures
on the plane sheet there will be a corresponding theorem concerning
figures on the curved sheet and vice versa, and similar methods of
proof may be employed in the two cases.

Though the objects about which we are reasoning in the two
cases are different, yet the logical processes are formally the same.

We can, however, go still further and consider the case where
the figures are accurately drawn (whatever that may mean) on
a plane sheet of india-rubber which is then stretched in any way.

In this case straight lines on the unstretched rubber would
become lines, straight or curved, on the stretched rubber and a
closed curve such as a circle would remain closed after the stretching.

Further, curves which intersected would still intersect and
curves which did not intersect would not intersect after the
stretching.

A point which lay inside a closed curve such as a circle, would
become a point inside a closed curve on the stretched rubber.

Again, a point which lay between two other points in a line of
some sort on the unstretched rubber would become a point between
two corresponding points on the corresponding line on the stretched
rubber.

PRELIMINARY CONSIDERATIONS 3

The distances between the points would of course have altered
according to our original standard and two lengths which were
originally equal might no longer be equal, but nevertheless certain
correspondences would still hold and could be traced between
theorems involving equality of lengths on the unstretched rubber
and theorems on the stretched rubber.

Perhaps the simplest way of seeing this is to introduce a system
of coordinates (say Cartesian coordinates) on the unstretched
rubber, by which any point of it would be represented by two
numbers.

If then we imagine the rubber to be stretched, the same pairs
of numbers could be taken to represent the same points of the
rubber after stretching as before. The axes would now, generally
speaking, become curved lines and the parallels to them would also
in general become curved lines.

The points equidistant from a given point on the unstretched
rubber would lie in a circle, and if the equation of this circle be

taken as ., , ,., 9

(x - a)2 + (y - b)z = r2,

then this equation would represent also some curve on the stretched
rubber. The radii of the circles would become some sort of lines all
passing through one point and intersecting the distorted circle.

We should in this way get lines which had been straight, curves
which had been circles, lengths which had been equal, etc., and we
could deal with these algebraically in the same way as we did with
the straight lines, circles and equal lengths on the unstretched
rubber.

We notice that the things which actually do remain permanent
are the particles of the rubber and certain features of their order.-,

If we consider the coordinate system we observe that, although
the axes and the parallels to them are in general no longer straight
after the stretching, yet as either set of parallel lines did not
intersect before stretching, so the corresponding lines do not
intersect after stretching and they preserve their original order.

We know however that, after a proper foundation has been
laid, any geometrical theorem may be proved by coordinate
methods and so it is evident that all reasoning which is done after
coordinates have once been introduced will apply equally in dealing
with certain other things than lines which are truly straight and
lengths which are truly equal.

1—2

4 TIME AND SPACE

Thus though the sheet of rubber may have originally been
plane, yet after stretching it may be curved in innumerable different
ways and yet there are certain features which remain invariant
throughout.

It is thus evident that although for purposes of mathematical
reasoning the actual straightness of lines or actual equality of
lengths in the ordinary sense of the terms is not essential, yet when
we wish to make use of geometry to describe the physical world
the meanings of "straightness" and "equality of length" are all-
important.

It is not sufficient that we should say that "there are such
things as straight lines," or that "there are lengths which are
equal," but it is necessary to have criteria by which we can say
(at least approximately) "here are points which lie in a straight
line" and "here is a length which is equal to yonder length."

The ruler and compasses give us rough standards of straightness
and equality of length in the sense in which these terms are used
in ordinary life, but, if we wish to go in for extreme accuracy,
other standards must be employed and we must get more precise
ideas as to what we really wish to convey when we make use of
such expressions.

Consider first the question of what we mean when we say that
two bodies are of equal length.

The ordinary method of comparing them is to make use of
a measuring rod which we regard as rigid; or an inextensible
tape line. But what do we mean by these words "rigid" and
"inextensible"'}

We find that it is by no means easy to say exactly what we do
mean.

Approximate rigidity and inextensibility are common enough
properties of solid bodies, but by the application of force all bodies
are found to be more or less elastic, while change of temperature
will also change the length of a rod compared with a parallel rod.

Again, if we wish to compare lengths which are not parallel,
the usual mode of procedure would be to turn a measuring rod
round from parallelism with the one length into parallelism with
the other.

The possibility then arises that during the motion the standard
may alter and give us results which indicate the lengths as equal
when in reality (whatever that may mean) they are different.

PRELIMINARY CONSIDERATIONS 5

Thus for example, if we wished to compare the lengths OA and
OB where A and B are, say, the extremities of the major and
minor axes of an ellipse whose centre
is O, and suppose we had an elastic
tape line which we place first with one
end at O and the other at B.

If then keeping the one end fixed
at 0 we move the other round the
ellipse we should apparently get the
same length for OA as for OB.

Now although this seems fan-
tastic, yet the famous experiment of
Michelson and Morley seemed to
show that just this sort of thing did
happen when a body was turned
round from a position such that its
length was parallel to the direction
of the earth's motion in its orbit into a position such that its length
was perpendicular to that direction.

The experiment, which was an optical one, consisted in dividing
a beam of light into two portions which travelled, the one in one
direction, and the other in a transverse direction, and were reflected
back again by mirrors.

If we adopt ordinary ideas for the moment and suppose
the light to be propagated with a velocity v through a medium
and the apparatus to move through that medium with a velocity
u, it is easy to calculate the time of the double journey for the two
portions of the beam.

For the case of a part of the beam which travels in the direction
of motion of the apparatus and back again the time of the double
journey is found to be 2uaA

where a1 is the distance between the point of the apparatus where
the beam divides and the corresponding reflector. For the case of
the transverse portion of the beam the time of the double journey

is found to be 2o2

t.> = — ,

where a2 is the distance between the point of the apparatus where
the beam divides and the other reflector.

6 TIME AND SPACE

Now it is possible to arrange things so that ^ = tz and this can
be done with extreme accuracy by means of the interference bands
which are produced.

We should then have

/ /S\i

ai = V l-(v a*'

giving

Thus av would be slightly less than a2.

It was found however that, when the apparatus was caused to
rotate at a uniform slow rate, and the times of the double journey
were equal for one position of the apparatus, then they were equal
for all positions. This seemed to indicate that the dimensions of
the apparatus in different directions changed as it rotated and the
view was put forward by Fitzgerald and Lorentz that a material
solid body contracted in the direction of its motion so that a sphere
moving through space with a velocity w became a spheroid whose
major and minor axes were in the ratio

1:

where v is the velocity of light.

It is clear that this once more raises the question as to the real
meaning of "equality of length" from which we started out.

Solid bodies apparently do not provide us with standards
sufficiently permanent for dealing with such problems.

But the subject of motion raises a number of other difficulties.

There is in particular the question of "absolute motion" and
whether this expression has any precise meaning.

The underlying idea of those who believe in "absolute motion"
is that, if we consider a definite point of space at any instant, then
that point preserves its identity at all other instants. The difficulty
of identifying a point of space at two different instants is freely
admitted, but for all that (so it is contended) the identity persists.

It was however noticed that, in the classical Newtonian
Mechanics, the equations of motion preserved the same form for
a system of bodies whose centre of inertia was in uniform motion
in a straight line as for a similar system whose centre of inertia
was "at rest," so that purely mechanical phenomena could not be
expected to show up any difference between the two cases.

PRELIMINARY CONSIDERATIONS 7

The question then naturally arose whether any difference could
be detected by optical or electrical means, but experiment failed
to show any.

Xextly it was pointed out by Larmor and Lorentz that the
electromagnetic equations could also be transformed by a linear
substitution so that they preserved the same form for a system
moving with uniform velocity as they had for a system "at rest."

In order to do this, however, a '"''local time" had to be intro-
duced.

We are all familiar with the use of "local time" on the earth's
surface, but the cases are different in one important respect.

The idea underlying the use of "local time" on the earth's
surface is simply that of having different names in different parts
of the world for what is regarded as the same instant. Thus noon
at Greenwich and noon at New York are both described as 12 o'clock
local time, although the instants referred to are clearly different.
On the other hand the use of chronometers in navigation is regarded
as a method of approximately identifying the same instant at
different parts of the earth.

But, as previously remarked, the "local time" used in trans-
forming the electromagnetic equations is of a different character
and events which are regarded as simultaneous according to one
"local time" would not be simultaneous, in general, when compared
by the "local time" of a system which was in motion with respect
to the first.

We might of course regard the one "local time" as the true
time and the other as a mathematical fiction, but there is no reason
known why we should select the one rather than the other, just
as there is no way of distinguishing a body "at rest" from one
moving uniformly in a straight line.

In fact it appears that, just as we have no method of distinguish-
ing the same point of space at two distinct instants of time, so we
cannot strictly identify the same instant of time at two distinct
points of space.

It is to be observed that though we started out by trying to
give a precise meaning to the idea of equality of length, in which
we seemed to be concerned only with space, yet in our attempt to
do so, we find difficulties with regard to time intruding themselves.

We can see, however, that even in our original use of compasses
the time element intrudes, since in comparing lengths by the use

8 TIME AND SPACE

of compasses, the compasses are moved and the idea of motion
involves that of time.

Also in the Michelson and Morley experiment, since light takes
a finite interval of time in getting from an object P to an object Q
and back again to P, we are introducing time relations in comparing
lengths.

The question now arises: suppose we imagine a flash of light
sent out at an instant A from a particle P to a distant particle
Q and arriving there at an instant B and suppose it reflected back
to P where it arrives at an instant C; how are we to identify the
instant B with any instant at P between A and C?

If we regard P as being "at rest" we might reasonably think
to identify B with the instant at P which is midway between A and
C, but this would imply that we had some means of measuring
intervals of time, and that brings us up against all the same sort
of difficulties which we encountered in trying to find a satisfactory
method of measuring space intervals.

On the other hand, if P be in uniform motion in the direction
PQ it would seem that B is not identical with the instant at P
which is midway between A and C.

In any case we do not know of any means of telling whether
P is "at rest" or not.

Having thus been led on from the consideration of spacial
relations to those of time we seem at first sight to have increased
our difficulties instead of solving them, but if we persevere in our
task we shall find that we have made an appreciable advance
towards solving our problem.

From the consideration of figures drawn on a sheet of rubber
which was afterwards stretched in any way, we were led to
recognise the importance of order in the study of the logic of
geometry, and since order also plays a part in time relations, it
seems worth while to consider order in time.

Now here we find an interesting and very important thing.

If I consider two distinct instants of which I am directly
conscious*, I notice that the one is after the other.

Noon to-day is afternoon yesterday and I cannot invert the order.

There is in fact what is called an asymmetrical relation between
the two instants, such that if B be after A, then A is not after B.

* The fact that I am directly conscious of the two instants is very important,
in view of later developments.

PRELIMINARY CONSIDERATIONS 9

If we consider two points or two particles in space, say P and
•Q, there is nothing analogous to this and we have no reason to
say that Q is after P rather than that P is after Q.

We might, of course, give them an order "by means of some
convention, but such convention would be quite arbitrary, whereas
in the case of the instants, it is a matter of fact and not of conven-
tion, quite independently of what words we may employ to express
that fact.

The simplest relation of order among points is a relation of
between which involves three terms instead of two.

This relation of between has been employed by various mathe-
maticians in investigating the foundations of geometry, but the
relation of equality of lengths then appears as something extraneous,
grafted on to the system.

The use of an asymmetrical relation such as after appears to
have great advantages over a relation such as between in constructing
a theory of order and I have found it possible, by means of such
a relation, to construct a system of geometry of space and time.
It might perhaps more correctly be described as a geometry of
time, of which spacial geometry forms a part.

In constructing this system it is necessary to modify certain
currently accepted notions, but the modifications required all
appear to be capable of justification and the structure, when
completed, will be found closely to resemble our ordinary con-
ceptions.

We shall regard an instant as a fundamental concept which,
for present purposes, it is unnecessary further to analyse, and shall
consider the relations of order among the instants of which I am
directly conscious.

Thus for such instants we find the following properties:

(1) If an instant B be after an instant A, then the instant A ^Cjift'f'
is not after the instant B, and is said to be before it.

(2) If A be any instant, I can conceive of an instant which is V /
after A and also of one which is before A.

(3) If an instant B be after an instant A, I can conceive of an
instant which is both after A and before B.

(4) If an instant B be after an instant A and an instant C be ^
after the instant B, the instant C is after the instant A.

(5) If an instant A be neither before nor after an instant B, the
instants A and B are identical.

10

TIME AND SPACE

The set of instants of which I am directly conscious have thus
got a linear order.

But now let us consider the fifth of these properties.

It might at first sight be supposed that it was a necessary
consequence of our ideas of before and after. That it is really
logically independent of the other properties may be shown by
the help of a geometrical illustration. This illustration is very
suggestive and we purpose to make further use of it, but the logic
of our theory is independent of the illustration.

Suppose we have a set of cones having their axes parallel and
having equal vertical angles, and further, suppose each cone to
terminate at the vertex, which is however to be regarded as a point
of the cone.

We shall call such a cone having its opening pointed upwards
an a cone, and one with the opening pointed downwards a ft cone.

Thus corresponding to any point of space there is an a cone of
the set having the point as vertex, and similarly there is a /3 cone
of the set having the point as vertex.

Now it is possible by using such cones and making a convention
with respect to the use of the words before
and after to set up a type of order of the
points of space.

For the purposes of this illustration we
shall make the convention that, if Av be
any point and al and /^ be the correspond-
ing a and /3 cones, then any point A2 will
be said to be after Al provided it is distinct
from AI and lies either on or inside the
cone «! and will be said to be before A1
provided it is distinct from A^ and lies
either on or inside the cone /^ .

It is easy to see that with this conven-
tion we have the following:

(1) If a point B be after a point A,
then the point A is not after the point B
and is said to be before it.

(2) If A be any point, there is a point which is after A and
also a point which is before A.

(3) If a point B be after a point A there is a point which is
both after A and before B.

Fig. 2

PRELIMINARY CONSIDERATIONS 11

(4) If a point B be after a point A and a point C be after the
point B, the point C is after the point ^4.

We cannot however assert that if a point A be neither before
nor a/fer a point B, that the points ^4 and B need be identical.

This is easily seen since the point B might lie in the region
outside both the a and ($ cones of A. (Fig. 2.)

This illustration shows that the fifth condition is logically
independent of the other four.

The type of order which we have illustrated by means of the
cones, we shall speak of as conical order, but the logical develop-
ment of the subject is independent of this illustration.

We may note however in passing that, if A and B be distinct
points one of which is neither before nor after the other, then there
are points which are after both A and B and also points which
are before both A and B.

This follows since in this case the a cones of A and B intersect,
as do also the ft cones of A and B.

It should further be noted that if we have any line straight or
curved in space, but whose tangent nowhere makes a greater
angle with the axes of the cones than their semi -vertical angle,
then if we confine our attention to the points of any one such line,
we can assert that : if a point A be neither before nor after a point
B, the points A and B are identical.

Thus provided we confine our attention to the points of such
a line, the whole five conditions are satisfied.

Returning now to the consideration of instants, we observed
that there was a difficulty in identifying the same instant at
different places.

The relations of before and after, however, enable us to say in
certain cases that instants at a distance are distinct. Thus if I can
send out any influence or material particle from a particle P at
the instant A so as to reach a distant particle Q at the instant
B then this is sufficient to show that B is after and therefore distinct
from A.

If now the influence or material particle be reflected back to
P and arrives there at the instant C, then C is after and therefore
distinct from B, while moreover, C is after A.

Now suppose the influence be a flash of light or other instanta-
neous electromagnetic disturbance and we appear to have reached
a limit.

12 TIME AND SPACE

We do not seem to be able to send out any influence or material
particle from P at any instant after A so as to arrive at Q at the
instant B, and we do not seem to be able to send out any influence
or material particle from Q at the instant B so as to arrive at P
before the instant C.

In fact the range of instants at P which are after A and before
C appear to be quite separated from the instant B so far as any
influence is concerned.

Now let us suppose that light has this property.

It may or may not be strictly true of light but, provided there
be some influence which has this property (and others which we
shall specify later), such influence will serve for the purp6se in
hand, and we shall, provisionally at any rate, ascribe it to light.

Now B could at most be identical with only one of the instants
at P, and such instant would require to be after A and before C,
but we have no means of saying which instant it is.

The other instants in this range would then all be either before
or after B.

But what now do we really mean when we say that one instant
is after another or one event after another?

If I at the instant A can produce any effect however slight at
the instant B, this is sufficient to imply that B is after A.

A present action of mine may produce some effect to-morrow,
but nothing which I may do now can have any effect on what
occurred yesterday.

It appears to me that we have here the essential features of
what we really mean when we use the word after, and that the
abstract power of a person at the instant A to produce an effect
at a distinct instant B is not merely a sufficient, but also a necessary
condition that B is after A.

If we accept this as the meaning of after it would then appear
that no instant at P which is after A and before C is either before
or after B.

We have already seen that the idea of an element being neither
before nor after another element, and yet distinct from it, involves
no logical absurdity, and so if we give up the attempt to identify
the instant B with any instant at P we get a logicallv consistent
view of things.

Thus according to the view here adopted there is no identity of
instants at different places at all.

PRELIMINARY CONSIDERATIONS 13

We may express the idea in this form: the present instant,
properly speaking, does not extend beyond here.

Thus there are instants at a distance before the present instant
and after it, and also instants neither before nor after it, but such
instants are to be regarded as being all quite distinct from the
present instant here.

Thus, according to the view here adopted, the only really simul-
taneous events are events which occur at the same place.

The theory which we desire to expound with regard to time and
space may now briefly be described as follows:

Taking the above view of instants and the relations of before
and after, we express in terms of these relations the conditions that
the set of instants should have a conical order of a certain type.
We then find that we have got a description not only of time but
also of space such as that with which we are already familiar.

In fact we may be said to analyze spacial relations in terms of
the time relations of before and after.

In first approaching this subject it is a great assistance to have
some concrete way of representing the facts to our minds even
though such representation may make use of some of the conceptions
which we are trying to analyze.

In doing so one must remember however that the justification
of our theory lies in the logical procedure and not in the representa-
tion.

Thus in trying to convey a general idea of what we are doing
we shall find it both convenient and suggestive to make use of our
mental images of cones, in the way already described, in order to
picture what we mean by conical order.

The idea of conical order is not at all dependent on this representa-
tion, but is built up by a rather lengthy piece of reasoning from
the relations of before and after.

The representation by means of cones may be compared to the
rough scaffolding used in the erection of a building, which is
removed when the building is complete and its component parts
in position.

We must, however, be certain that the building is not supported
by the scaffolding, or it will not be able to stand alone.

In order to make sure of this in our theory, great care has to
be taken, and, for details on this matter, I must refer readers to
my larger work.

14 TIME AND SPACE

Moreover, the representation by means of cones gives only a
three-dimensional conical order, whereas the conical order with
which we are really concerned is a four-dimensional one.

The representation also introduces a sort of distortion, but
this need not trouble us when we deal only with descriptive
features.

Now in ordinary mathematical physics we are accustomed to
localize an instantaneous event by means of four numbers x, y, z, t.
Of these numbers as, y and z are called spacial coordinates while
t is referred to as the "time."

But now having come to regard all instants at different places
as distinct, we regard these four numbers as really representing
four coordinates of an instant.

The coordinate t, however, has different before and after relations
from those associated with the other three coordinates x, y, z,
which are made clear by the conception of conical order.

In order to avoid confusion therefore, we shall speak of the
former not as "time" but as a t coordinate.

We are not yet, however, in a position to introduce coordinates
except for "scaffolding" purposes.

Neither again are we at liberty to speak of "velocity" except
for scaffolding purposes until we have defined the meaning of the
word.

Moreover, in the actual proof of theorems, we must not employ
the ideas of equality of lengths or angles until these ideas are seen
to be definable in terms of before and after relations.

We may, however, make use of such terms in the "scaffolding"
which is mere poetry and rhetoric.

Let us therefore first consider this pictorial representation in
which we have to confine ourselves to three coordinates instead of
four, which we shall take to be x, y and t, and shall regard as
rectangular.

Now by taking suitable units we may regard the "velocity of
light" as unity and under these circumstances if we imagine a
flash of light starting from the position x = a, y = b, t = c, the
rays of light would be represented by the generators of the upper
half of the cone whose equation is

(x - a)2 + (y - &)2 - (t - c)2 = 0,
which we take as the a cone corresponding to (a, b, c).

PRELIMINARY CONSIDERATIONS 15

The lower portion of this same locus constitutes the ft cone of
<a, b, c).

The point (a, b, c) itself is regarded as belonging to both the
a and ft cones.

The successive positions of a material particle would be repre-
sented by some line straight or curved, but since it appears that
a particle of matter never quite attains to the "velocity of light"
the tangent to this curve would make an angle with the axis of
/ which is always less than 45° : the semi-vertical angle of the cones

The successive positions of a particle which remains at rest
with respect to the system of axes would be represented by a
straight line parallel to, or coincident with, the axis of t.

The successive positions of a particle which remains in uniform
motion with respect to the system of axes would also be represented
by a straight line, but one inclined to the axis of t.

The successive positions of an accelerated particle would be
represented by a curved line.

The set of instants of which any one individual is directly
conscious would also be represented by some line straight or curved,
whose tangent always makes an angle with the axis of t less than
the semi-vertical angle of the cones.

We thus see that for the set of instants of which any one
individual is directly conscious, or the set of instants which any
one particle occupies, we can assert that: if an instant A be neither
before nor after an instant B, the instants A and B are identical.

We cannot, however, assert this of the instants of which two
individuals are directly conscious, or which two distinct and
separate particles occupy.

It may be that an instant of which I am directly conscious is
neither before nor after some instant of which you are directly
conscious, but they are not identical, and our illustration shows
that this involves no logical contradiction.

It is to be noted, however, that if A and B be two distinct
instants, one of which is neither before nor after the other, then
since the a cones intersect and also the ft cones, there are instants
which are after both A and B and also instants which are before
both A and B, so that we may both speak of to-morrow or of
yesterday, though strictly speaking we have no common present.

Thus instead of regarding ourselves as, so to speak, swimming
along in an ocean of space (as we usually do), we are to think of

16 TIME AND SPACE

ourselves rather as swimming along in an ocean of time, while
spacial relations are to be regarded as the manifestation of the fact
that the elements of time form a system in conical order: a conception
which may be analyzed in terms of the relations of after and before.

Having thus given a sort of vista of the promised land, we must
next give some account of the rather toilsome journey entailed in
getting there.

CONICAL ORDER

IN building up the idea of conical order in terms of the relations
of before and after, we make use of certain postulates involving
these relations.

These postulates do not involve any idea of measurement but
are of a purely descriptive character.

If there are any physical facts corresponding to the postulates,
then we shall be able to describe such facts in terms of the relations
of before and after.

In the formal development of the theory we shall speak of
elements instead of instants and of a and /3 sub-sets instead of
a and /3 cones.

The a and fi sub-sets have, however, yet to be denned.

When we wish to form a concrete picture of these we shall
frequently make use of the cones, since by doing so we get sugges-
tions as to suitable postulates, and also as to methods of procedure
to prove theorems.

However, we supposed the cones to have their axes parallel
and to have equal vertical angles, and neither the idea of cone, of
parallel, of axis, of angle, or of equal have been analyzed in terms
of before and after, and must therefore be excluded in denning a
and (3 sub-sets.

The relations of before and after being converse asymmetrical
relations, either may be defined in terms of the other.

It is a matter of indifference which we take as fundamental,
but in the following account we shall start with the relation of
after.

Most of the postulates consist of two parts marked (a) and (b)
in which the relations before and after are interchanged.

In some, however, such as those numbered I, III and IV, the
one part follows from the other as a direct consequence of the

CONICAL ORDER 17

mutual relations of after and before, while in others such as V,
these relations are involved symmetrically.

We shall now proceed with an account of the formal develop-
ment of the theory.

We shall suppose that we have a set of elements and that
certain of these elements stand in a relation to certain other
elements of the set which we denote by saying that one element is
after another.

The first four postulates are the equivalents of the first four
characteristics which we observed as belonging to the set of
instants of which any one individual is directly conscious.

We shall re-state them as follows :

POSTULATE I. If an element B be after an element A,
then the element A is not after the element B.

Definition. If an element B be after an element A, then the
element A will be said to be before the element B.

POSTULATE II. (a) If A be any element, there is at least
one element which is after A.

(b) If A be any element, there is at least one element
which is before A.

POSTULATE III. If an element B be after an element A,
and if an element C be after the element B, the element C
is after the element A.

POSTULATE IV. If an element B be after an element A,
there is at least one element which is both after A and
before B.

The next postulate is one to admit of the existence of pairs of
instants, of which not more than one of a pair can be in the direct
consciousness of any single individual; or in other words, of the
existence of instants which, in respect of any one individual, are,
as we say, elsewhere. It is as follows :

POSTULATE V. If A be any element, there is at least one
other element distinct from A, which is neither before nor
after A.

In our illustration by means of cones an element distinct from
A. and neither before nor after it, is represented by a point outside
both the a and (3 cones of A.

E. 2

18 TIME AND SPACE

If B be such a point then as we have also seen, the a cones of
A and B intersect as do the /3 cones of A and B.

Now we wish to express in terms of before and after a character-
istic property of a point of intersection.

If we take X as a point on the locus of intersection of the
a cones of A and B and we take the plane through A, B and X,
we observe in the first place that X is after both A and B.

Further X is before all other points in the plane which are after
both A and B.

This is not, however, the case if we go outside this plane, for
if we take a second point X' on the locus of intersection, then X
is not before X', although X' is after both A and B.

We can, however, easily see that though we are not at liberty
to say that X is before, we can assert that X is not after any other
point which is after both A and B, and this gives us the property
we have been searching for.

CONICAL ORDER 19

Thus we may express our next postulate as follows:

POSTULATE VI. (a) If A and B be two distinct elements,
one of which is neither before nor after the other, there is
at least one element which is after both A and B, but is
not after any other element which is after both A and B.

(6) If A and B be two distinct elements, one of which
is neither after nor before the other, there is at least one
element which is before both A and B, but is not before
any other element which is before both A and B.

This last postulate, although somewhat complicated in form, is
extremely important as it enables us to give a definition of the
a. and ft sub-sets.

In fact reverting to our illustration, a point X which bears the
relation postulated in Post. VI (a) to two points A and B, one of
which is neither before nor after the other, must lie in the a cones of
both A and B.

Further, if X lies in the a cone of A and is distinct from it,
there exist points, such as B, which in general we may call Y.

As we wish to include A itself in the a cone or sub-set, we
mention it explicitly.

Thus we get the following definitions of the sub-sets :

Definition, (a) If A be any element of the set, then an element
X will be said to be a member of the a sub-set of A provided X is
either identical with A, or else provided there exists at least one
element Y distinct from A and neither before nor after A and such
that X is after both A and Y but is not after any other element
which is after both A and Y.

(b) If A be any element of the set, then an element X will
be said to be a member of the ft sub-set of A provided X is either
identical with A, or else provided there exists at least one element
Y distinct from A and neither after nor before A and such that
X is before both A and Y but is not before any other element which
is before both A and Y.

We must next express some further properties of the a and ft
sub-sets which obviously hold in the case of our cones.

We shall denote by av and /^ the sub-sets corresponding to an
element Al, and by «2 and /32 those corresponding to an element
Az, etc.

2—2

20 TIME AND SPACE

POSTULATE VII. (a) If Aa and'A^ be elements and if A2
be a member of ax then Ax is a member of jS2.

(b) If A1 and A2 be elements and if A2 be a member of
jSl then Aj is a member of a2.

POSTULATE VIII. (a) If A! be any element and A2 be any
other element in al} there is at least one other element
distinct from A2 which is a member both of a: and of ct2.

(b) If Aj be any element and A2 be any other element
in /?!, there is at least one other element distinct from A,
which is a member both of jSl and of fl2.

By the help of the above postulates we can prove a number of
theorems.

Thus our first theorem is that: if Al be any element and A2 be
any other element in «15 then any element A3 which is both after
AI and before Az must be a member both of ax and /?2.

This is easily seen to hold in the case of the cones.

We can also show that there are elements in the a sub-set of
any element which are distinct and neither before nor after one
another; and similarly in the {$ sub-set.

The next step is to define what corresponds to a generator of
a complete cone, or a and f$ taken together.

Now reverting to our illustration, we notice that if one point
lies on the a cone of another and is distinct from it, then the two
cones touch along a generator.

We accordingly introduce the following:

Definition. If Al be any element and A2 be any other element
in «! , the optical line A^A^ is defined as the aggregate of all elements
which lie either

(1) both in «j and «2,

or (2) both in ax and /?2,

or (3) both in j8x and j$z.

Before we can prove the chief properties of what we have called
an optical line it is necessary to introduce another postulate.

Taking our usual illustration, the thing which represents an
optical line is a generator of the combined a and ft cones.

Now if we consider any such generator there are points which,
while not being points of the generator itself, are before points of

CONICAL ORDER 21

it and similarly there are points which are not points of the generator
but are^after points of it.

We are able from the postulates already given to prove the
existence of elements having corresponding properties.

Now if we consider a point which is not a point of a given
generator but is before some point of it, it is easy to see that the
a cone of the first point has one single point in common with the
generator.

We have also a corresponding result if we consider a point
which is not a point of the generator but after some point of it, in
which case the /3 cone of the first point has one single point in
common with the generator.

We accordingly introduce the following postulate :

POSTULATE IX. (a) If a be an optical line and A1 be any
element which is not in the optical line but before some
element of it, there is one single element which is an
element both of the optical line a and the sub-set al.

(b) If a be an optical line and Al be any element which
is not in the optical line but after some element of it, there
is one single element which is an element both of the optical
line a and the sub-set /?x .

We can now prove a number of theorems which are important
in the logical development, such as the existence of elements
having certain specific properties.

We can also prove that of any two elements of an optical line
the one lies in the a sub-set of the other and is therefore after it.

It can also be shown that there are an infinite number of
elements in an optical line and that they are in a linear order as
distinguished from a conical one.

It can also be proved that any two elements of an optical line
determine that optical line.

Further, we have the interesting result that if an element A1
be before an element of an optical line and also after an element of
it, then At is itself an element of the optical line.

Having thus defined an optical line and proved some of its
chief properties, the next obvious thing to do is to try to define
the representatives of lines which are not generators of a and ft
cones, but this is not so easy. The method by which I succeeded

22 TIME AND SPACE

in doing this was by taking the intersections of two planes of a
certain type.

Making use of our illustration once more, we notice that if we
take the combined a and J3 cones of any point and take a straight
line passing through the vertex, such straight line may either (i) be
a generator of the cone (corresponding to an optical line) ; or (ii) may
fall inside the cone; or (iii) may fall outside the cone.

Again, if we take a plane through the vertex it may either
(i) be a tangent plane to the cone; or (ii) may cut the cone in two
generators; or (iii) may have no real point in common with the
cone except the vertex.

Only the second type of plane contains all three types of line,
but the method of defining such a plane offers some difficulties.

We may note, however, that through any point of such a plane
there are two and only two lines of the type of what we have called
optical lines which lie entirely in the plane. In fact there are two
distinct parallel sets of these lines lying in the plane.

Now the method employed in defining a plane in most work on
the foundations of geometry is to take a triangle and to define the
plane as the aggregate of all points of all lines which intersect two
side lines of the triangle in distinct points ; or some equivalent of this.

This method is not, however, open to us, since the only lines
we have denned cannot form a triangle.

Another method which suggests itself is to take two parallel
lines and to define the plane as the aggregate of all points of all
of a system of lines which intersect the first two.

This seems more hopeful, but the difficulty remains of defining
the parallelism of the two lines to begin with.

The ordinary definition of parallel lines implies the prior notion
of a plane in which the lines lie.

We thus seem to be up against a formidable difficulty.

But now we notice that only planes of types (i) and (ii) contain
lines of the character of optical lines.

A plane of type (i) contains only one set of parallel lines of
this character, while a plane of type (ii) contains two sets of such
parallel lines.

Now a little consideration will show us that if a and b be two
parallel lines of the type corresponding to optical lines lying in
a plane of type (i), then no point of a is either before or after any
point of b.

CONICAL ORDER 23

If the lines a and b lie in a plane of type (ii), then either each
point of a is after a point of b, or else each point of a is before a
point of b. If finally we take two lines a and b of this type which
are oblique to one another and do not intersect, it is not difficult
to see that there are some points of a which are after points of
b and others which are before points of b and these two sets are
separated by a single point of a which is neither before nor after
any point of b.

These considerations give us the clue to a new postulate as
follows :

POSTULATE X. (a) If a be an optical line and if A be any
element not in the optical line but before some element of
it, there is one single optical line containing A and such
that each element of it is before an element of a.

(b) If a be an optical line and if A be any element not
in the optical line but after some element of it, there is one
single optical line containing A and such that each element
of it is after an element of a.

We can easily show that if each element of one optical line be
before an element of another distinct optical line, the two optical
lines cannot have an element in common.

Similarly, if each element of the one be after an element of the
other.

We can also prove that if each element of an optical line a be
before an element of another optical line b, then through each
element of a there is one single optical line which intersects b; and
similarly if each element of a be after an element of b.

We are not, however, as yet in the position to carry out the
plan which we outlined for defining the types of line other than
optical lines.

It will be remembered that it was proposed to define a certain
type of plane and to define these other types of line by means of
the intersection of two such planes.

To have two planes intersecting it would be necessary to have
more than two dimensions, and all the postulates which we have
hitherto introduced may be represented in one plane.

It is easy to introduce a postulate which gives at least a three-
dimensional character to our system, and this might have been
done by a slight alteration of Postulate VI (a) and (6).

24 TIME AND SPACE

If we substitute in it for the words "there is at least one element
which'1'' the words: there are at least tivo elements either of which,
we should have had what we wanted; but leaving Postulate VI in
its original form we can express the same thing by a new postulate
as follows:

POSTULATE XI. (a) If Ax and A2 be two distinct elements,
one of which is neither before nor after the other, and X
be an element which is a member both of ax and a2, then
there is at least one other element distinct from X which
is a member both of ax and a2.

(b) If Aj and A2 be two distinct elements, one of which
is neither after nor before the other, and X be an element
•which is a member both of jS^ and /?2, then there is at
least one other element distinct from X which is a member
both of/?j and/?2.

It is easily seen that if A± and Az be two distinct elements such
that the one is neither before nor after the other and if X and X' be
two distinct elements lying both in ax and in az, then X is neither
before nor after X'.

Similarly if X and X' lie in both & and j82.

It is not difficult to prove now that if A be any element there
are at least three distinct optical lines containing A.

It will be observed, however, from our illustration that not
more than two of the lines corresponding to optical lines which
pass through any point lie in any one plane.

Now we have already observed in our illustration that if a and
b be two of the lines corresponding to optical lines which neither
intersect nor are parallel, there is one single point of a which is
neither before nor after any point of b, and we now wish to make
use of the corresponding property of optical lines in order to
investigate their parallelism. We accordingly introduce another
postulate as follows :

POSTULATE XII. (a) If a be an optical line and A be any
element not in the optical line but before some element of
it, then each optical line through A, except the one which
intersects a and the one of which each element is before
an element of a, has one single element which is neither
before nor after any element of a.

CONICAL ORDER 25

(b) If a be an optical line and A be any element not in
the optical line but after some element of it, then each
optical line through A, except the one which intersects a
and the one of which each element is after an element of
a, has one single element which is neither after nor before
any element of a.

We are now able to prove that: if each element of an optical
line a be after an element of a distinct optical line b, then each
element of b is before an element of a and vice versa.

This is not obvious, as might perhaps at first sight appear.

We can also prove now that : if a be an optical line and if A±
be any element which is neither before nor after any element of
a, there is one single optical line containing A± and such that no
element of it is either before or after any element of a.

We have already seen how this is represented in our illustration.

We can now define the parallelism of optical lines as follows :

Definitions. An optical line a will be said to be parallel to a
second distinct optical line b when either

(1) each element of a is after an element of b,
or (2) each element of a is before an element of b,
or (3) no element of a is either before or after any element of b.

In case (1) a will be said to be an after-parallel of b.

In case (2) a will be said to be a before-parallel of b.

In case (3) a will be said to be a neutral-parallel of b.

We can show that if a be parallel to b, then b is parallel to a.

Also we can show that if a be an optical line and A be any
element not in the optical line, then there is one single optical line
parallel to a and containing A.

Further, we are able to prove a number of theorems which
combined together give us the general result that:

If two distinct optical lines a and b are each parallel to a third
optical line c, then the optical lines a and b are parallel one to
another.

We may now give the following definition :

Definition. If a and b be any pair of distinct optical lines one
of which is an after-parallel of the other, then the aggregate of all
elements of all optical lines which intersect both a and b will be
called an acceleration plane.

As we have alreadv seen in our illustration what we have called

26 TIME AND SPACE

an acceleration plane is represented by a plane through the vertex
of a cone of the system intersecting the cone in two distinct
generators.

We have given it the name acceleration plane because it might
be supposed to be determined by the acceleration of a particle,
but for the present we shall simply take it as a name.

We can easily prove that there are an infinite number of
acceleration planes which contain any given optical line.

We can now introduce a new postulate.

POSTULATE XIII. If two distinct acceleration planes
have two elements in common, then any other acceleration
plane containing these two elements contains all elements
common to the two first mentioned acceleration planes.

We can now prove that if a and b be two distinct optical lines
and if a be an after-parallel of b, then if c and d be two other distinct
optical lines intersecting both a and b, one of the optical lines c, d
is an after-parallel of the other.

We can also show that an acceleration plane contains two and
only two optical lines which pass through any element of it and
these form two parallel systems.

Further, if an acceleration plane contain an optical line a and
an element A^ which does not lie in the optical line, then Al is
either before or after an element of a.

We can, moreover, show that there are an infinite number of
acceleration planes containing any pair of elements, whether the
one be after the other or neither before nor after it.

We can also show that if two or more distinct acceleration
planes contain an optical line, there is no other element which
they have in common which does not lie in the optical line.

We may now introduce the following definitions :

Definitions. If two acceleration planes contain two elements in
common, then the aggregate of all elements common to the two
acceleration planes will be called a general line.

If two acceleration planes contain two elements in common, of
which one is after the other but does not lie in the same optical
line with it, then the aggregate of all elements common to the two
acceleration planes will be called an inertia line.

If two acceleration planes contain two elements in common, of
which one is neither before nor after the other, then the aggregate

CONICAL OKDER 27

of all elements common to the two acceleration planes will be
called a separation line.

The name inertia line is employed because it appears to represent
the set of instants which an imaccelerated particle occupies ; while
the name separation line is used because particles which occupy
distinct elements of such a line would be separated, as we say, "in
space."

In the formal development of our subject, however, we shall
simply regard these as names.

If we take our usual illustration and consider a complete cone,
then an inertia line would be represented by a straight line passing
through the vertex and Iving inside the cone, while a separation
line would be represented by one passing through the vertex but
lying outside the cone.

In order to investigate the properties of an inertia line we
introduce a fresh postulate whose representation in our illustration
is obvious.

POSTULATE XIV. (a) If a be any inertia line and Al be
any element of the set, then there is one single element
common to the inertia line a and the sub-set a.^.

(b) If a be any inertia line and A1 be any element of the
set, then there is one single element common to the inertia
line a and the sub-set j3l.

This should be compared with Post. IX.

It is to be observed that any element of the set is after certain
elements of any inertia line and before certain others, whereas if
any element be after one element of an optical line and before
another element of it, it must itself be an element of the optical
line. Moreover, it follows from Post. XII that there are elements
of the set which are neither before nor after any element of a given
optical line.

We are now in a position to prove a number of the characteristic
properties of an inertia line.

Thus we can show that an inertia line in any acceleration plane
has one single element in common with each optical line in the
acceleration plane.

We can also show that of any two distinct elements of an inertia
line one is after the other and so no inertia line can be a separation
line.

28 TIME AND SPACE

We can in fact show that the elements of an inertia line have
the whole five characteristic properties which we mentioned as
belonging to the set of instants of which any one individual is
directly conscious.

This does not mean, however, that the latter set of instants
are confined to an inertia line.

In order to investigate the properties of a separation line we
introduce the following postulate :

POSTULATE XV. If two general lines, one of which is a
separation line and the other is not, lie in the same ac-
celeration plane, then they have an element in common.

We further make the following definition:

Definition. An element in an acceleration plane will be said to
be between a pair of parallel optical lines in the acceleration plane
if it be after an element of the one optical line and before an element
of the other and does not lie in either optical line.

We can now show that if A± and A2 be any two distinct elements
of a separation line, there is at least one other element of the
separation line which lies between a pair of parallel optical lines
through AI and A2 respectively in an acceleration plane containing
the separation line.

We can also show that any such element lies between the other
pair of parallel optical lines in the acceleration plane which pass
through A1 and A2.

In fact by making use of parallel optical lines we can assign
an order to the elements of a separation line, in so far as a
particular acceleration plane is concerned, but we are not yet in
a position to show that this order is independent of the particular
acceleration plane in which the separation line is regarded as
lying.

Another interesting result which we may mention is that if
A be an element of an optical line a and if B be an element which
is neither before nor after any element of a, then no element of the
separation line AB, with the exception of A, is either before or
after any element of a.

It is clear that in this case the optical line a and the element
B cannot lie in one acceleration plane. In fact they lie in another
type of plane which is represented in our illustration by a tangent
plane to a cone.

CONICAL ORDER 29

Now if A!, Az and A3 be three distinct elements which do not
all lie in one general line, this last result shows that they either
may or may not lie in one acceleration plane.

If they do lie in one acceleration plane they must determine the
acceleration plane, since should they lie in a second one they would
have to lie in one general line, contrary to hypothesis.

It is possible at this stage to give certain criteria by which we
can say whether or no a set of three elements does lie in one
acceleration plane.

These will be found in my larger work.

We can now define the parallelism of acceleration planes.

If we call the optical lines in an acceleration plane the generators
of the acceleration plane this definition is as follows :

Definition. If an acceleration plane have its two sets of generators
respectively parallel to the two sets of generators of another
distinct acceleration plane, then the two acceleration planes will be
said to be parallel to one another.

It is easy to see that if P be an acceleration plane and A be
any element outside it, then there is one single acceleration plane
containing A and parallel to P, and further this acceleration plane
can have no element in common with P.

Further, two distinct acceleration planes which are parallel to
the same acceleration plane are parallel to one another.

There is no distinction of different sorts of parallelism of
acceleration planes as there is for optical lines.

We can now prove that: if an acceleration plane P have one
element in common with each of a pair of parallel acceleration
planes Q and R then, if P have a second element in common with
Q it has also a second element in common with R.

Now we have already noticed that if two distinct optical lines
intersect a pair of optical lines, one of which is an after-parallel
of the other, then of the two first-mentioned optical lines one is
an after-parallel of the other.

We shall accordingly state the following definition:

Definition. If two distinct optical lines intersect a pair of
optical lines, one of which is an after-parallel of the other, then
the four optical lines will be said to form an optical parallelogram.

It is evident that an optical parallelogram lies in an acceleration
plane.

We can give obvious definitions to corners, opposite, or adjacent*

30 TIME AND SPACE

We can also give obvious definitions to diagonal lines of an
optical parallelogram, and it is easy to see that one of these must
be an inertia line and the other a separation line.

We shall now give a new postulate which gives a relation
between certain inertia lines and certain separation lines.

POSTULATE XVI. If two optical parallelograms lie in the
same acceleration plane, then if their diagonal lines of one
kind do not intersect, their diagonal lines of the other kind
do not intersect.

It is now possible to show that if a be any general line in an
acceleration plane P and Al be any element of the acceleration
plane which is not in the general line, then there is one single
general line through A± in the acceleration plane which does not
intersect a, and further, this general line must be of the same type
as a.

We can also show that if two acceleration planes P and Q have
a general line a in common, and if Av be any element which does
not lie either in P or Q, then the acceleration planes through A±
parallel to P and Q, respectively, have a general line in common.

This last result enables us to give a definition of the parallelism
of a pair of general lines whether these lie in one acceleration plane
or not.

It may be expressed as follows :

Definition. If a be a general line, and A be any element which
does not lie in it, and if two acceleration planes R and S through
A are parallel respectively to two others P and Q containing a,
then the general line which R and S have in common is said to
be parallel to a.

It is easy to see that this covers the case of the parallelism of
optical lines which is the only case of the parallelism of general
lines which we had hitherto defined.

We are able to show now that: if a be a general line, and A±
be any element which does not lie in it, there is one single general
line containing A^ and parallel to a.

We can also show that if two distinct general lines are each
parallel to a third, then they are parallel to one another.

If a and b be any pair of parallel general lines, it is easy to see
that they must be general lines of the same kind, for we have found
in the course of our work that two parallel general lines in one

CONICAL ORDER 31

acceleration plane must be of the same kind, and by two applications
of this result it follows that if a and b do not lie in one acceleration
plane they must also be of the same kind.

A number of other theorems concerning parallel general lines
may now be proved which are analogous to corresponding theorems
about parallel straight lines in ordinary geometry.

These are important in the strict logical development of our
subject, but as we are here only concerned with giving an outline of
the course of procedure, I must again refer readers to my larger work.

Definition. The element of intersection of the diagonal lines of
an optical parallelogram will be called the centre of the optical
parallelogram.

We can now prove that : if two distinct elements A and 0 be
taken in an inertia or separation line in a given acceleration plane,
then there is one single optical parallelogram in the acceleration
plane having 0 as centre and A as one of its corners.

We can also give demonstrations of some other very important
theorems concerning optical parallelograms.

We can show in the first place that : if two optical parallelograms
have two opposite corners in common, then they have a common
centre.

For the optical parallelograms to be distinct, it is clear that
they must lie in different acceleration planes.

Again, we can show that: if two optical parallelograms have
two adjacent corners in common, then optical lines through the
centres of the optical parallelograms and intersecting their common
side line intersect it in the same element.

Here the two optical parallelograms either may, or may not,
lie in the same acceleration plane.

We can now introduce the following definitions :

Definition. If A and B be two distinct elements lying in an
inertia line or in a separation line, then the centre of an optical
parallelogram of which A and B are a pair of opposite corners will
be spoken of as the mean of the elements A and B.

Definition. If A and B be two distinct elements lying in an
optical line, then an optical line through the centre of an optical
parallelogram of which A and B are a pair of adjacent corners and
intersecting the optical line AB, intersects it in an element which
will be spoken of as the mean of the elements A and B.

The above-mentioned theorems show that the mean of the

32

TIME AND SPACE

elements A and B is a definite element independent of the particular
optical parallelogram used to define it. This marks the first stage
on the way to introducing the ideas of equality of lengths and of
measurement.

We have now to introduce a new postulate
which bears a sort of analogy to the well-known
axiom of Archimedes, but which, unlike the
latter, does not contain any reference to con-
gruence.

Before doing so, however, it is necessary to
go into certain points in connexion with it.

If a and b be any two distinct inertia lines,
and AQ be any element in a which is not an
element of intersection with b, then from Post.
XIV (a) it follows that there is one single ele-
ment common to the inertia line b and the a
sub-set of AQ.

Call this element B0.

Then B0 is distinct from A0 and cannot be
an element of intersection of the two inertia
lines, for if it were A0 and B0 would lie both
in an inertia line and an optical line which is
impossible.

Further, there cannot be an element of inter-
section of the inertia lines lying after A0 and
before B0, for then, as was pointed out on page
21, such element would require to lie in the
optical line A0B0 and so again we should have
two elements lying both in an optical line and
an inertia line which is impossible.

Thus any element of intersection of the two
inertia lines, if such an element exists, must lie
either before A0 or after B0.

Again, from Post. XIV (a) it follows that
there is one single element, say Alt common to
the inertia line a and the a sub-set of B0, and again, A± cannot
be an element of intersection of the inertia lines.

Further, any such element, if it exists, must lie either before A0
or after A±.

Proceeding again in the same way, there is one single element,

Fig. 4

CONICAL ORDER 33

say Bj_, common to the inertia line b and the a sub-set of Alt
and one single element A2 common to the inertia line a and the
« sub-set of Bl , and so on.

We thus get an infinite series of elements A0, Alt A2, A3 ... in
the inertia line a and another infinite series of elements B0, J515
B2 , B3 . . . in the inertia line b.

An element of intersection of the two inertia lines, if ^such an
element exists, must lie either before A0 or after An , where n is any
finite integer whatever.

This process will be spoken of as taking steps along the inertia
line a with respect to the inertia line b.

The passing from A0 to Al is the first step, the passing from
Ai to A2 the second, and so on.

If ,Y be an element in a which is after A0 and before An but not
before An_-±, then the element X will be said to be surpassed from
A0 in n steps taken along a with respect to b.

If C be an element of intersection of the two inertia lines, and
if C be after AQ, it is evident from what we have said that C cannot
be surpassed from A0 in any finite number of steps.

We can now introduce our new postulate.

POSTULATE XVII. If A0 and A^, be two elements of an
inertia line a such that A, is after A0, and if b be a second
inertia line which does not intersect a either in A0, A.x or
any element both after A0 and before Ax, then Ax may be
surpassed in a finite number of steps taken from A0 along
a with respect to b.

It follows directly from Post. XVII that if the two inertia lines
do not intersect at all, then Ax may always be surpassed in a finite
number of steps.

There is also a (b) form of this postulate, but it is not inde-
pendent, and in fact is proved in Theorem 64 of my larger work.

The primary use to which we put this postulate is to prove
certain theorems which entail repeated constructions of some
particular type.

Without the postulate we should have no guarantee that certain
elements came within the scope of our proofs.

Making use of results obtained in this way, we are now able to
prove the following important theorem:

If A, B and C be three elements in a separation line, and if

34 TIME AND SPACE

B be between a pair of parallel optical lines through A and C in
an acceleration plane containing the separation line, then B is also
between a pair of parallel optical lines through A and C in any
other acceleration plane containing the separation line.

This is a result which clearly holds in our ordinary illustration,
but hitherto we had not been able to demonstrate it from our
postulates.

The importance of the theorem lies in the fact that we can now
assign a definite order to the elements of a separation line, which
is independent of a particular acceleration plane.

We accordingly introduce the following definition:
Definition. If three distinct elements lie in a general line, and
if one of them lies between a pair of parallel optical lines through
the other two in an acceleration plane containing the general line,
then the element which is between the parallel optical lines will
be said to be linearly between the other two elements.

The above definition is so framed as to apply to all three types
of general line and is therefore more complicated than it need be
if we were dealing only with optical or inertia lines.

For the case of elements lying in either of these types of general
line, one element is linearly between two other elements if it be
after the one and before the other.

In the case of elements lying in a separation line, however, no
one is either before or after another, and so we have to fall back
on our definition involving parallel optical lines.

The distinction between the three cases is interesting, and is
made clear if we make use of our usual illustration.

Thus if three elements A, B and C lie in a general line a, and
if B be linearly between A and C, then in case a be an inertia line
we must either have B after A, and C after B, or else B after C
and A after B, and similarly when a is an optical line.

If a be an inertia line, and B be after A, and C after B, then
B will be before elements of both optical lines through C, and after
elements of both optical lines through A in any acceleration plane
containing a.

If a be an optical line, and B be after A, and C after B, then,
apart from a itself, there is only one optical line through any
element of a in any acceleration plane containing a, and so we
should have B before an element of the optical line through C and
after an element of the parallel optical line through A.

CONICAL ORDER 35

If a be a separation line, however, we should have B before an
element of one of the optical lines through C, and after an element
of the parallel optical line through A, and also after an element of
the second optical line through C, and before an element of the
parallel optical line through A in any acceleration plane containing a.

Inertia line Optical line Separation line

Fig. 5

Now Peano has given a number of axioms of the straight line
in ordinary geometry in terms of the relation of between and the
corresponding properties may now be shown to hold for the general
line in our geometry in terms of the relation of linearly between as
we have denned it.

We can also prove two theorems which correspond to two axioms
given by Peano for points in a plane.

These theorems are as follows :

If A, B and C be three elements in an acceleration plane which
do not all lie in one general line, and if D be an element linearly
between A and B, while E is an element linearly between B and C,
there exists an element which lies both linearly between A and E
and linearly between C and Z>.

If A, B and C be three elements in an acceleration plane which
do not all lie in one general line, and if D be an element linearly
between A and B, while F is an element linearly between C and
D, there exists an element, say E, which is linearly between B
and C, and such that F is linearly between A and E.

The proofs of these theorems require the consideration of a
number of different cases and are therefore somewhat tedious.

It is to be noted that these theorems have as yet only been
proved for elements in an acceleration plane.

In addition to the analogues of the various axioms of Peano

3—2

36 TIME AND SPACE

concerning the ordering of points in a line or plane, we have also
seen that if a be any general line in an acceleration plane, and A
be any element of the acceleration plane which does not lie in
a, then there is one single general line passing through A and lying
in the acceleration plane which does not intersect a.

This corresponds to the Euclidean axiom of parallels.

An acceleration plane, however, differs from a Euclidean plane,
since there are three types of general line in the former and only
one type of straight line in the latter.

The congruence properties of the two are also quite different
as will be seen later.

We shall now introduce the following definition:

Definition. If two parallel general lines in an acceleration plane
be both intersected by another pair of parallel general lines, then
the four general lines will be said to form a general parallelogram
in the acceleration plane.

It should be noted that we shall afterwards extend the term
general parallelogram to figures in other types of plane.

We can now prove that if we have a general parallelogram in
an acceleration plane, then:

(1) The two diagonal lines intersect in an element which is the
mean of either pair of opposite corners.

(2) A general line through the element of intersection of the
diagonal lines and parallel to either pair of side lines, intersects
either of the other side lines in an element which is the mean of
the pair of corners through which that side line passes.

We can also prove the theorem that: if A, B and C be three
elements in an acceleration plane which do not all lie in one
general line, and if D be the mean of A and B, then a general line
through D parallel to BC intersects AC in an element which is
the mean of A and C.

Again, we can show that if A0 and An be two distinct elements
in a general line a, we can always find n— 1 elements AlfAz, ... An_^
in a (where n — 1 is any integer), such that :

A1 is the mean of A0 and Az,
Az is the mean of Av and A3,

_i is the mean of An_2 and An .

CONICAL ORDER 37

We have now to take up the study of the type of plane which in
our illustration is represented by a tangent plane to a complete cone.

The first step consists in proving certain theorems in connexion
with neutral-parallel optical lines. Thus we can prove the following:

If A be any element in an optical line a, and A' be any element
in a neutral-parallel optical line a', then if B be a second element
in a the general line through B parallel to A A' intersects a' in an
element, say B'.

Further, if B be after A, then B' is after A', while if B be before
A then B' is before A'.

Again, we can show that under the above conditions there is
only one general line through B and intersecting a' which does
not intersect the general line A A': namely, the parallel to A A'.

Again, if a and b be two neutral-parallel optical lines, and if
one general line intersects a in A and b in B, while a second general
line intersects a in A and b in B', then an optical line through
any element of AB and parallel to a or b intersects A'B'.

We can now show that if a and b be two neutral-parallel optical
lines, and if c and d be any two non-parallel separation lines which
intersect both a and b, then the aggregate consisting of all the
elements in c and in all separation lines intersecting a and b which
are parallel to c must be identical with the aggregate consisting
of all the elements in d and in all separation lines intersecting a
and b which are parallel to d.

This prepares the way for the following definition :

Definition. The aggregate of all elements of all mutually
parallel separation lines which intersect two neutral-parallel
optical lines will be called an optical plane.

It is evident that through any element of an optical plane there
is one single optical line lying in the optical plane.

In analogy with the case of an acceleration plane, an optical
line which lies in an optical plane will be called a generator of the
optical plane.

It is now easy to show that if two distinct elements of a general
line lie in an optical plane, then every element of the general line
lies in the optical plane.

However, only optical and separation lines lie in an optical
plane, and in this important respect it differs from an acceleration
plane which contains also inertia lines.

We can easily prove that if e be a general line in an optical

38 TIME AND SPACE

plane, and A be any element of the optical plane which does not
lie in e, then there is one single general line through A in the
optical plane which does not intersect e.

This is the parallel to e through A and the result given corre-
sponds to the axiom of parallels in Euclidean geometry.

We can also easily show that Peano's axioms of order in a
plane hold for an optical plane making use of the already defined
meaning of linearly between.

We can also show that if A, B and C be three elements in an
optical plane which do not all lie in one general line, and if D be
the mean of A and B, then a general line through D parallel to
BC intersects AC in an element which is the mean of A and C.

We can further prove a number of theorems in connexion with
an optical plane analogous to those proved for an acceleration
plane, and so bring our knowledge of the former up to a level
with our knowledge of the latter.

It is easy to show that if three elements Alf A2, A3 which do
not lie in one general line lie in one optical plane, they determine
the optical plane containing them and we are able to establish
criteria by which we can say whether or no three elements do lie
in an optical plane.

If we return to the consideration of our illustration there is,
as already pointed out, a third type of plane besides those which
represent acceleration and optical planes.

We have next to consider certain theorems in our geometry
preparatory to the investigation of a type of plane analogous to
a plane in our illustration which passes through the vertex of a
cone, but has no other real point in common with the cone.

The first theorem is as follows :

If two optical parallelograms have a pair of opposite corners
in common lying in an inertia line, then their separation diagonal
lines are such that no element of the one is either before or after
any element of the other.

If c and e be two such separation diagonal lines, then, as we
have already seen, they have a common element, namely: the
centre of the two optical parallelograms.

Further, any general line which intersects c and e in distinct
elements must itself be a separation line.

It is easy to show that the separation lines c and e cannot
both lie either in one acceleration plane or in one optical plane.

CONICAL ORDER 39

Before we can carry out the investigation of this third type of
plane, we have to introduce a new postulate.

If A be any element, and a be an inertia line not containing A,
while B is the element common to a and the a sub-set of A, then
we shall speak of B as the first element in a which is after A.

Similarly, if C be the element common to a and the j8 sub-set
of A, we shall speak of C as the last element in a which is before A.

The neAV postulate is as follows :

POSTULATE XVIII. If a, b and c be three parallel inertia

lines which do not all lie in one acceleration plane and Al

be an element in a and if

BJ be the first element in b which is after Aj ,
GJ be the first element in c which is after Aj,
B2 be the first element in b which is after C1 ,
Co be the first element in c which is after Bx,

then the first element in a which is after B2 and the first

element in a which is after C2 are identical.

From the physical standpoint this postulate looks to be among
the simplest, and corresponds to the presumed optical fact that if
we have three particles P, Q and R which are unaccelerated, at
rest with respect to one another, and at the corners of a triangle,
then if a flash of light starting from P goes directly round from
P to Q to R and back again to P, and if another flash starting from
P goes directly round from P to R to Q and back again to P, then
if the two flashes start out simultaneously from P they return
simultaneously to P.

The postulate is not, however, quite so simple as it appears,
since it implies that we already know just what we mean when
we say that the three particles are at rest with respect to one
another and are unaccelerated.

We have, however, defined the meaning of inertia lines and
their parallelism so that we have already overcome this difficulty.

If we take our ordinary illustration and represent the inertia
lines by lines parallel to the axis of t, then we can see at once that
the postulate holds for such lines, and it is not difficult to prove
that it also holds if the lines representing the inertia lines are not
parallel to the axis of t.

There is a (b) form of this postulate in which the word last is

40 TIME AND SPACE

substituted for the word first, and the word before for the word
after, but it is not independent and can readily be proved from the
form given.

Definition. An inertia line and a separation line which are
diagonal lines of the same optical parallelogram will be said to be
conjugate to one another.

It is evident that if an inertia line and a separation line are
conjugate they lie in one acceleration plane and intersect one
another.

It is also evident that if A be an element lying in an inertia or
separation line a in an acceleration plane P, then there is only
one separation or inertia line through A and lying in P which is
conjugate to a; since if two optical parallelograms lie in P and
have a as a common diagonal line, then their other diagonal lines
do not intersect (Post. XVI).

From this it also follows that if two intersecting separation
lines b and c be both conjugate to the same inertia line a, then
a, b and c cannot lie in the same acceleration plane, and we shall
have a and b in one acceleration plane, say P, while a and c lie in
another, say Q.

If 0 be the element of intersection of b and c, then 0 must lie
both in P and Q and therefore in the inertia line a.

If A! be an element in a distinct from 0 there is one optical
parallelogram in the acceleration plane P having 0 as centre and
A! as one of its corners.

If

parallelogram in Q also having
corners and therefore having the same centre 0.

The separation lines b and c will be the separation diagonal
lines of the optical parallelograms in P and Q respectively, and so it
follows that no element of b is either before or after any element of c.

By considerations similar to the above, we can see that if two
intersecting inertia lines b and c be both conjugate to the same
separation line a, then a and b must lie in one acceleration plane,
while a and c lie in another distinct acceleration plane.

Further, if 0 be the element of intersection of b and c, then
0 lies in a.

In this case, however, since b and c are two intersecting inertia
lines they must lie in one acceleration plane which must be distinct
from both the others.

CONICAL ORDER 41

Again, it is clear that if a be an inertia or separation line lying
in an acceleration plane P with a separation or inertia line b which
is conjugate to a, then any general line c lying in P and parallel
to b is also conjugate to a.

Also conversely, it is clear that if a be an inertia or separation
line lying in an acceleration plane P with two distinct separation
or inertia lines b and c which are each conjugate to a, then b and
c must be parallel to one another.

By the help of Post. XVIII we are able to prove some other
very important theorems regarding conjugacy such as the following:

If an inertia line c be conjugate to two intersecting separation
lines d and e, then if A be any element of d and B be any distinct
element of e, the general line AB is conjugate to a set of inertia
lines which are parallel to c.

Also we can show that: if two inertia lines b and c intersect
in an element Av and are both conjugate to a separation line a,
then a is conjugate to every inertia line in the acceleration plane
containing b and c which passes through the element Al.

In the course of proving this result it is also shown that no
element of a with the exception of A± is either before or after any
element of either of the generators of the acceleration plane con-
taining b and c which pass through A{.

We can also prove the theorems :

If b and c be any two intersecting inertia lines, there is at least
one separation line which is conjugate to both b and c.

Also if b and c be any two intersecting separation lines such
that no element of the one is either before or after any element of
the other, there is at least one inertia line which is conjugate to
both b and c.

Further, if an inertia line a be conjugate to a separation line
b, and if an inertia line a' be parallel to a while a separation line
b' is parallel to b, and if a' and b' intersect one another, then a' is
conjugate to b'.

Having proved these various theorems on conjugacy we are
now able to go on with the proof of the existence of the third type
of plane suggested by our illustration.

The first step is to prove the following theorem :

If a be a separation line, and B be any element which is not
an element of a, and is neither before nor after any element of a,
while c is a general line passing through B and parallel to a, then

42 TIME AND SPACE

if A be any element of a, while C is an element of c distinct from
B, a general line through C parallel to BA will intersect a.

We can easily show that no element of c is either before or after
any element of a.

We can next show that: if A and B be two elements lying
respectively in two parallel separation lines a and b which are such
that no element of the one is either before or after any element of
the other, and if A' be a second and distinct element in a, there
is only one general line through A' and intersecting b which does
not intersect the general line AB.

This is in fact the parallel to AB through A'.

We are ultimately able to prove that if a and b be two parallel
separation lines such that no element of the one is either before or
after any element of the other, and if c and d be any two non-
parallel separation lines intersecting both a and b, then the
aggregate consisting of all the elements in c and in all separation
lines intersecting a and b which are parallel to c must be identical
with the aggregate consisting of all the elements in d and in all
separation lines intersecting a and b which are parallel to d.

Definition. If a and b be two parallel separation lines such that
no element of the one is either before or after any element of the
other, then the aggregate of all elements of all mutually parallel
separation lines which intersect both a and b will be called a
separation plane.

We can now show that if any two distinct elements of a general
line lie in a separation plane, then every element of the general
line lies in the separation plane.

It is easily seen that no element of a separation plane is either
before or after any other element of it.

We can also prove various properties of a separation plane
such as the analogues of Peano's axioms and the parallel axiom,
and can bring our state of knowledge of a separation plane up to
the level of our knowledge of acceleration and optical planes.

We can also obtain a criterion to show under what circumstances
three elements which do not lie in one general line lie in a separation
plane, and can prove that any three elements which do not all lie in
one general line must lie either in an acceleration plane, an optical
plane, or a separation plane.

We can also now introduce the term general plane as a common
designation for all three types and may define it as follows :

CONICAL ORDER 43

Definition. If a and b be any two intersecting general lines,
then the aggregate of all elements of the general line b, and of all
general lines parallel to b which intersect a, will be called a general
plane.

The axioms of Peano are now seen to hold for a general plane
as docs also the equivalent of the axiom of parallels.

It is not difficult to see that: if a and b be two intersecting
general lines lying in a general plane P, and if through any element
not lying in P two general lines a' and b' be taken respectively
parallel to a and b, then if P' be the general plane determined by
a' and b', the two general planes P and P' can have no element in
common.

We can also show that if a' and b' intersect in 0' there is a
general line through 0' and lying in P' which is parallel to any
general line in P.

We have already given a definition of the parallelism of
acceleration planes and are now in a position to give a definition
of the parallelism of general planes which will include that of
acceleration planes as a special case.

Definition. If P be a general plane, and if through any element
A outside P two general lines be taken respectively parallel to
two intersecting general lines in P, then the two general lines
through A determine a general plane which will be said to be
parallel to P.

We can readily see that : through any element outside a general
plane P, there is one single general plane parallel to P.

We can also see that this general plane must be of the same kind
as P.

We can further show that two distinct general planes which
are parallel to a third general plane are parallel to one another.

If a pair of parallel general lines be both intersected by another
pair of parallel general lines, then the four general lines will form
a general parallelogram either in an acceleration plane, an optical
plane, or a separation plane.

We can now prove that: if two general parallelograms have a
pair of adjacent corners in common their remaining corners either
lie in one general line or else form the corners of another general
parallelogram.

This result is important when we come to develop the theory
of congruence.

44 TIME AND SPACE

We have next to consider some theorems relating to the con-
jugacy of inertia and separation lines which lead on to the more
general conception of the normality of general lines, of which
indeed conjugacy is a special case.

We can show that if an inertia line a be conjugate to two
separation lines intersecting in the element 0 and lying in a separa-
tion plane P, then a is also conjugate to every separation line lying
in P and passing through 0.

Further, if b be any such separation line, then a and b lie in
an acceleration plane, say Q, and we can prove that there is one and
only one separation line, say c, lying in P and passing through 0
which is conjugate to every inertia line in Q which passes through 0.

Moreover, if R be the acceleration plane containing a and c,
then we can also show that b is conjugate to every inertia line in
R which passes through O.

It is thus seen that there is a reciprocal relation between the
separation lines b and c.

We have now to introduce a new postulate.

All the postulates which have hitherto been introduced may
be represented by ordinary geometric figures not involving more
than three dimensions, but we have now to introduce one which
we cannot thus represent and which therefore gives our geometry
a sort of four-dimensional character. It is as follows :

POSTULATE XIX. If P be any optical plane there is at
least one element which is neither before nor after any
element of P.

If we consider our usual illustration, we have seen that an
optical plane is represented by a tangent plane to one of our cones
and any point in the three-dimensional space which contains these
cones is either before or after some point of such a plane.

A point which actually lies in the plane will be before certain
points of it and after certain others, but a point outside the plane
cannot bear this double relation and must be either before or else
after certain points of the plane.

We cannot represent in three dimensions the case of an element
such as that whose existence is asserted in Post. XIX.

We may compare this with an analogous feature of an optical
line in an acceleration plane.

In that case, any element in the acceleration plane is either

CONICAL ORDER 45

before or after some element of the optical line, and if we take the
special case of an element of the optical line itself, it is before
certain elements of the optical line and after certain others.

The only way in which we can get an element which is neither
before nor after any element of the optical line is if we go outside
the acceleration plane.

It will be observed that the analogy is very close.

It is easy to show by the help of Post. XIX that : there are at
least two distinct optical planes containing any optical line, and
this might be taken as an alternative form of the postulate.

In proving a number of theorems from this stage on, our usual
illustration fails us and we have to get along without it.

However, in many of the theorems which we require to con-
sider, we can get a great part of the construction into three
dimensions and the remaining portion is easy, but in any case the
practice which we have had in proving theorems by means of the
relations of before and after enables us to treat four-dimensional
theorems without much extra difficulty.

The first theorem which we prove by means of this postulate
is as follows:

If b be any separation line, and 0 be any element in it, there
are at least two acceleration planes containing O, and such that
b is conjugate to every inertia line in each of them which passes
through 0.

Another important result which we reach is that we may have
an acceleration plane and a separation plane having only one
element in common, and such that each inertia line through the
common element in the former is conjugate to every separation
line through it in the latter.

In ordinary three-dimensional geometry, of course, we cannot
have two planes with only one point in common.

Again, the following theorem may be proved:

If two distinct acceleration planes P and P' have a separation
line 6 in common, and if another separation line c intersecting b
in the element 0 be conjugate to every inertia line in P which
passes through O, then if c be conjugate to one inertia line in P'
which passes through O, it is conjugate to every inertia line in
P' which passes through 0.

46 TIME AND SPACE

NORMALITY OF GENERAL LINES HAVING A
COMMON ELEMENT

WE are now in a position to define what we mean when we say
that a general line a is normal to a general line b which has an
element in common with it.

Since a and b are not always general lines of the same kind, it
is difficult to give any simple definition which will include all cases;
but the introduction of the word "normal" is justified by the
simplification which is thereby brought about in the statement of
many theorems.

Moreover, once we have introduced coordinates, the condition
of the normality of general lines is the same in all cases.

Only one case will be found to be strictly analogous to the
normality of intersecting straight lines in ordinary geometry;
namely, the case of two separation lines.

The other cases are so different from our ordinary ideas of
lines "aZ right angles'''' that we do not propose to use this expression
in connexion with them.

Thus for instance any optical line is to be regarded as being
"normal to itself" and the use of the words "at right angles"
would in this case clearly be an abuse of language.

The extension of the idea of normality from the cases of general
lines having a common element to the cases of general lines which
have not a common element is, however, quite analogous to the
corresponding extension in ordinary geometry and will be made
subsequently.

We are at present only concerned with the cases of general
lines having a common element and shall naturally include among
these that of an optical line being "normal" to itself.

Thus the complete definition of the normality of general lines
having a common element is to be taken as consisting of the following
four particular definitions A, B, C and D.

Definition A. An optical line will be said to be normal to itself.

Definition B. If an optical line a and a separation line b have
an element 0 in common, and if no element of b with the exception
of O be either before or after any element of a, then b will be said
to be normal to a and a will be said to be normal to b.

Definition C. If an inertia line a and a separation line b be

NORMALITY OF GENERAL LINES 47

conjugate one to the other, then a will be said to be normal to
b and b will be said to be normal to a.

Definition D. A separation line a having an element 0 in
common with a separation line b will be said to be normal to b
provided an acceleration plane P exists containing b, and such
that every inertia line in P which passes through 0 is conjugate
to a.

In this last case since there is one single inertia line in P which
passes through 0 and is conjugate to b, it is evident that a and b lie
in a separation plane.

Further, the result mentioned immediately prior to the intro-
duction of Post. XIX shows that in this case the relation between
a and b is a reciprocal one, so that b satisfies the definition of being
normal to a provided a is normal to b.

It is clear from our definitions that for general lines which
have an element in common we may have:

(i) A separation line normal to an inertia line, an optical line,
or a separation line.

(ii) An optical line normal only to an optical line or a separation
line.

(iii) An inertia line normal only to a separation line.

Again, if a be a separation line, and b a general line intersecting
a and normal to it, then:

(i) If b be an inertia line, a and b lie in one acceleration plane.

(ii) If b be an optical line, a and b lie in one optical plane.

(iii) If & be a separation line, a and b lie in one separation plane.

If P be a separation plane, and if b be any separation line in
P, and O be any element in 6, we can easily show that there is one,
and only one, separation line in P and passing through 0 which is
normal to b.

Again, if instead P be an acceleration plane, and if b be any
general line in P, and 0 be any element in b, then we know that
if b be either an inertia or a separation line there is one and only
one general line in P and passing through 0 which is conjugate
and therefore normal to b.

Also from our definitions if b be an optical line there is still one
and only one general line in P and passing through 0 which is
normal to b : namely, b itself.

Thus we have the following general result :

// P be either a separation plane or an acceleration plane, and

48 TIME AND SPACE

if b be any general line in P and 0 be any element in b, then there
is one and only one general line lying in P and passing through
0 which is normal to b.

Now we have seen that if a separation line a be normal to
a separation line b having an element in common with it, then
a and b lie in a separation plane.

Thus two intersecting separation lines in an optical plane
cannot be normal one to another.

Any separation line, however, which lies in an optical plane is
normal to every optical line in the optical plane, since no element
of the separation line except the element of intersection is either
before or after any element of any optical line in the optical plane.

Since there is one, and only one, optical line which passes
through any element of an optical plane and lies in the optical
plane, we have the following result :

If P be an optical plane, and if b be any separation line in P,
and 0 be any element in b, then there is one, and only one, general
line lying in P and passing through 0 which is normal to b.

If on the other hand b be an optical line lying in P, then every
general line in P which passes through 0 (including b itself ) is normal
to b.

There is another very important theorem concerning the
normality of general lines which is as follows :

// b and c be two distinct general lines having an element O in
common, and if a general line a passing through 0 be normal to both
b and c, then a is normal to every general line which passes through O
and lies in the general plane containing b and c.

There are a number of special cases of this theorem which have
already been mentioned, and some others which have not been
referred to.

An enumeration of the different cases will be found in my larger
work.

We can now prove that if b and c be two general lines inter-
secting in an element 0 and such that the one is normal to the
other, and if they are respectively parallel to two other general
lines b' and c' intersecting in an element 0', then of these latter
two general lines the one is normal to the other.

We can now give the following definition :

Definition. A general line b will be said to be normal to a general
line c' which has no element in common with it, provided that

NORMALITY OF GENERAL LINES 49

a general line b' taken through any element of c' parallel to b is
normal to c' in the sense already denned.

Since any optical line is normal to itself, it follows from the
above definition that any two parallel optical lines are to be
regarded as normal to one another.

Definition. A general line a will be said to be normal to a general
plane P provided a be normal to every general line in P.

It is evident that if a general line a be normal to two inter-
secting general lines in a general plane P, then a will be normal to P.

In case P be an optical plane it is clear that, according to the
above definition, any generator of P is normal to P.

This is the only case in which a general line can be normal to
a general plane which contains it.

In no other case can a general line which is normal to a general
plane have more than one element in common with the latter.

It has already been pointed out that we may have an accelera-
tion plane and a separation plane having only one element in
common and such that each inertia line through the common
element in the former is conjugate to every separation line through
it in the latter.

It is evident now that we have here two general planes which
are so related that any general line in the one is normal to any
general line in the other.

In ordinary three-dimensional geometry two planes cannot be
so related, and when we speak of one plane being normal to another,
the normality is not of this complete character.

We shall therefore introduce the following definition :

Definition. If two general planes be so related that every
general line in the one is normal to every general line in the other,
the two general planes will be said to be completely normal to one
another.

We are now able to prove that if P be any general plane and
0 be any element in it, there is at least one general plane passing
through 0 and completely normal to P.

In fact if P be an acceleration plane the general plane com-
pletely normal to it whose existence we prove is a separation plane,
if P be a separation plane the completely normal general plane is
an acceleration plane, while if P be an optical plane, the completely
normal general plane is an optical plane.

We can, however, go further and instead of taking 0 as any

50 TIME AND SPACE

element in P we can prove a similar result if O be any element
whatever.

Again, let O be any element and let S be any separation plane
passing through 0, while P is an acceleration plane passing through
0 and completely normal to S.

Let a be any separation line in S which passes through O, and
let b be the one single separation line in S and passing through
0 which is normal to a.

Let c be any separation line passing through O and lying in P,
and let d be the one single inertia line in P and passing through
0 which is normal to c.

Then both c and d are normal to both a and b and so we have
the three separation lines a, b and c all passing through O and each
of them normal to the other two ; while in addition to these we have
the inertia line d also passing through 0 and normal to all three.

This marks an important stage in the development of our
theory as it suggests the possibility of setting up a system of
coordinate axes one of which axes is of a different character from
the remaining three.

Various other theorems concerning normality which are im-
portant in the logical development may now be proved and among
these we mention the following:

If three general lines a, b and c have an element 0 in common,
there is at least one general line passing through 0 which is normal
to all three.

We are now in a position to take another important step in
the development of our subject by defining three/olds.

In defining general lines and general planes, it was found
necessary to define particular types separately, before any general
definition could be given.

This difficult v does not occur in the case of general threefolds.

Definition. If a general line a and a general plane P intersect,
then the aggregate of all elements of P and of all general planes
parallel to P which intersect a will be called a general threefold.

It will be found that, just as there are three types of general
line and three types of general plane, so there are three types of
general threefold.

The distinction will be made hereafter, but there are many
properties possessed in common by all three types and of which
general proofs may be given.

NORMALITY OF GENERAL LINES 51

Thus we can show that if two distinct elements of a general
line lie in a general threefold, then every element of the general
line lies in the general threefold.

Also if a general plane have three distinct elements in common
with a general threefold, and if these three elements do not lie in
one general line, then every element of the general plane lies in
the general threefold.

Further, if a general line b lies in a general threefold W, and if
A be any element lying in W but not in b, then the general line
through A parallel to b also lies in W.

Also if a general plane P lies in a general threefold W, and if
A be any element lying in W but not in P, then the general plane
through A parallel to P also lies in W.

Moreover, we can show that if a general threefold W be deter-
mined by a general plane P and a general line a which intersects
P, then if Q be any general plane lying in W and if b be any general
line lying in W and intersecting Q, the general plane Q and the
general line b also determine the same general threefold W.

It follows from this that any four distinct elements which do
not all lie in one general plane determine a general threefold
containing them ; as do also any three distinct general lines having
a common element and not all lying in one general plane.

We can also prove that if two distinct general planes P and
Q lie in a general threefold W, then if P and Q have one element
in common they have a second element in common and have there-
fore a general line in common.

Moreover, if P and Q have no element in common but both lie
in W then they must be parallel.

Now we have already seen that we can have a separation plane
S and an acceleration plane P having an element 0 in common
and which are completely normal to one another.

We have also seen that in this case P and S cannot have a
second element in common.

It follows that P and S cannot lie in one general threefold.

Now let ax and a2 be any two distinct general lines lying in
P and passing through 0.

Then S and % determine a general threefold, say Wlt while
S and a2 determine a general threefold, saw Wz.

Now W^ and W2 must be distinct, for if W2 were identical with W^
then W± would contain both ax and «2 and would therefore con tain P.

52 TIME AND SPACE

But Wi contains S and so this is impossible.

Thus Wj and Wz are distinct general threefolds, each of which
contains the separation plane S.

Since there are an infinite number of general lines lying in
P and passing through 0, it follows that there are an infinite
number of general threefolds which all contain any separation
plane S.

Similarly, there are an infinite number of general threefolds
which all contain any acceleration plane P.

Without Post. XIX or some equivalent, we cannot from our
remaining postulates show that there is more than one general
threefold; for the proof of the existence of an acceleration plane
which is completely normal to a separation plane depends upon
Post. XIX.

We can now prove that: if a, b and c be any three distinct
general lines having an element O in common but not all lying in
one general plane, and if a general line d also passing through 0 be
normal to «, b and c, then d is normal to every general line in the
general threefold containing a, b and c.

Definition. A general line which is normal to every general line
in a general threefold will be said to be normal to the general three-
fold.

We are now able to prove the existence of the three different
types of general threefold and to give criteria by which we can tell
in which kind of general threefold a given set of four elements (not
all in one general plane) must lie.

The three different types may be defined as follows :

Definition. If a separation line a intersects a separation plane
S and is normal to it, then the aggregate of all elements of S and
of all separation planes parallel to S which intersect a will be called
a separation threefold.

Definition. If an optical line a intersects a separation plane
S and is normal to it, then the aggregate of all elements of S and
of all separation planes parallel to S which intersect a will be called
an optical threefold.

Definition. If an inertia line a intersects a separation plane S
and is normal to it, then the aggregate of all elements of S and of
all separation planes parallel to S which intersect a will be called
a rotation threefold.

As regards the characteristic properties of these different types ;

NORMALITY OF GENERAL LINES 53

no element of a separation threefold is either before or after any
other element of it, and so the only type of general lines which
it contains are separation lines and the only type of general planes
are separation planes.

As regards an optical threefold, through any element of it there
is one single optical line lying in the optical threefold and all these
optical lines are neutrally parallel to one another. These will be
spoken of as generators.

An optical threefold contains separation lines and optical lines
but no other type of general line, and it contains separation planes
and optical planes but no other type of general plane.

As regards a rotation threefold, it contains all three types of
general line and all three types of general plane.

Through any element of it there are an infinite number of
optical lines, which we shall call generators, lying in the rotation
threefold and in fact all the postulates which we introduced prior
to Post. XIX hold for the set of elements contained in a rotation
threefold without going outside that rotation threefold.

This is clearly not true for separation or optical threefolds.

We are now in a position to introduce a new postulate which
limits the number of dimensions of our set of elements.

POSTULATE XX. If W be any optical threefold, then any
element of the set must be either before or after some
element of W.

This postulate should be compared with Post. XIX and the
difference noted.

In Post. XIX the existence of an element neither before nor
after any element of an optical plane is asserted, while in Post. XX
the existence of an element neither before nor after any element of
an optical threefold is denied.

As in the case of an optical line and an optical plane, so too in
the case of an optical threefold, if any element be after one element
of it and before another it must itself be an element of the optical
threefold.

We are now able to prove that : if P be any general plane and
O be any element of the set, there is one, and only one, general
plane passing through 0 and completely normal to P.

We had previously shown that there was at least one such
general plane and now we can show that it is unique.

54 TIME AND SPACE

We can also prove that :

(1) If P be any acceleration or separation plane and 0 be any
element outside it, then the general plane through 0 and com-
pletely normal to P has one single element in common with P.

(2) If P be an optical plane and 0 be any element outside it,
the optical plane through O and completely normal to P has an
optical line in common with P if 0 be neither before nor after any
element of P, but has no element in common with P if O be either
before or after any element of P.

We can further prove that: if W be any general threefold and
0 be any element of the set, there is one, and only one, general
line passing through 0 and normal to W.

Also if a be any general line and 0 be any element of the set
there is one, and only one, general threefold passing through 0 and
normal to a.

Further, we can show that : if a be a general line and 0 be any
element in it while W is a general threefold passing through 0 and
normal to a, then

(1) If a be an inertia line, W is a separation threefold.

(2) If a be a separation line, W is & rotation threefold.

(3) If a be an optical line, W is an optical threefold containing a.
We can now show that: if W7 be a general threefold and A be

any element outside it, then any general line through A is either
parallel to a general line in W or else has one single element in
common with W ' .

Since a separation threefold contains only separation lines,
while an optical threefold contains only separation lines and a
system of parallel optical lines, it follows from the last result that :
every inertia line and every optical line intersects every separation
threefold, while every inertia line and every optical line which is
not parallel to a generator of an optical threefold intersects the
optical threefold.

Again, we can show that if W be a general threefold and P be
a general plane which does not lie in W, then if P has one element
in common with W, it has a general line in common with W.

Further, if W^ and Wz be two distinct general threefolds having
an element A in common, then they have a general plane in common.

We can also prove that: if W be a general threefold and 0 be
any element outside it, and if through 0 there pass three general
lines a, b and c which do not all lie in one general plane and which

NORMALITY OF GENERAL LINES 55

are respectively parallel to three general lines in W, then a, b and
c determine a general threefold W, such that every general line
in W is parallel to a general line in W.

We accordingly introduce the following definition :

Definition. If W be a general threefold, and if through any
element A outside W three general lines be taken not all lying in
one general plane but respectively parallel to three general lines in
W, then the general lines through A determine a general threefold
which will be said to be parallel to W.

It is clear that a general threefold can only be parallel to
another general threefold if they are both of the same type.

It is also easy to see that if A be any element outside a general
threefold W, there is only one general threefold passing through
A and parallel to W.

Further, if two distinct general threefolds are both parallel to
the same general threefold they are parallel to one another.

There are various cases of partial normality of general planes
to general planes and general threefolds, and of general threefolds
to general threefolds considered in my larger work, but which we
do not propose to go into here.

There is one very important theorem we shall now enunciate
which is required in the treatment of the theory of congruence:
the subject we propose next to take up. It is as follows:

If A, J5, C, D be the corners of an optical parallelogram (AC
being the inertia diagonal line), and if A, B', C, D' be the corners
of a second optical parallelogram, while A', B', C', D' are the
corners of a third optical parallelogram whose diagonal line A'C'
is conjugate to BD, then A', B, C', D will be the corners of a fourth
optical parallelogram.

The importance of this theorem lies in the fact that it enables
us to show the unique character of the relations of certain elements
in respect of congruence.

56 TIME AND SPACE

THEORY OF CONGRUENCE

WE are now in a position to consider the problems of congruence
and measurement in our system of geometry.

The first point to be considered is the congruence of pairs of
elements and we shall find that there are several cases which have
to be considered separately.

Two distinct elements A and B will be spoken of briefly as
a pair and will be denoted by the symbols (A, B) or (B, A).

The order in which the letters are written will be taken advantage
of in order to symbolize a certain correspondence between the
elements of pairs, as we shall shortly explain.

Since any two distinct elements determine a general line
containing them, there will always be one general line associated
with any given pair, but different pairs will be associated with the
same general line.

If we set up a correspondence between the elements of a pair
(A, B) and a pair (C, D) we might either take C to correspond to
A and D to B, or else take D to correspond to A and C to B.
The first of these might be symbolized briefly by :

(A, B) corresponds to (C, D),
or (B, A) corresponds to (D, C).

The second might be symbolized by:

(A, B) corresponds to (D, C),
or (B, A) corresponds to (C, D).

If we consider the case of pairs which have a common element,
say (A, B) and (A, C), and if

(A, B) corresponds to (A, C),

then the element A corresponds to itself and will be said to be
latent.

Now the congruence of pairs is a correspondence which can be
set up in a certain way between certain pairs lying in general lines
of the same type.

The correspondence is set up in virtue of certain similarities
of relationship.

In dealing with this subject, it will be found convenient to have
a systematic notation for optical parallelograms, so that we may
be able to distinguish how the different comers are related.

THEORY OF CONGRUENCE 57

If A, B, C, D be the corners of an optical parallelogram we
shall use the notation A BCD when we wish to signify that the
corners A and D lie in the inertia diagonal line and that A is before
D, while B and C lie in the separation diagonal line so that the
one is neither before nor after the other.

If O be the centre of the optical parallelogram ABCD, it is
obvious that O will be after A and before D.

A pair (A, B) will be spoken of as an optical pair, an inertia
pair or a separation pair according as AB is an optical, an inertia,
or a separation line.

We shall first give a definition of the congruence of inertia pairs
having a latent element.

Definition. If A1[BCDl and A2BCD2 be optical parallelograms
having the common pair of opposite corners B and C and the
common centre O, then the inertia pair (0, DJ will be said to be
congruent to the inertia pair (0, Z>2).

This will be written

Similarly the inertia pair (O, Aj) will be said to be congruent
to the inertia pair (O, A2).

If (0, Dj) be any inertia pair and a be any inertia line inter-
secting OZ>j in O, then the above definition enables us to show that
there is one and only one element, say X, in a which is distinct
from O and such that :

(0, D1)( = )(0,X).

The unique character of the element X is proved by means of
the theorem stated at the end of the last section.

Again, if (O, Dx), (O, D2) and (O, D3) be inertia pairs such that:

(0,DJ(*)(Q.D&

and (0,Da)(=)<0,D8),

it is easy to show that :

(0,DJ(=)(Otp9).

In order to see this we have only to remember that whether
the inertia lines OZ)15 OD2, OD3 all lie in one acceleration plane or
in one rotation threefold, there must be at least one separation line
passing through 0 and normal to all three.

Thus for inertia pairs having a latent element the relation of
congruence is a transitive relation.

58 TIME AND SPACE

We shall next consider the congruence of separation pairs
having a latent element.

This case differs somewhat from the one we have considered.

While two intersecting inertia lines always lie in an acceleration
plane, two intersecting separation lines may lie either in a separa-
tion plane, an optical plane, or an acceleration plane.

An inertia line can only be conjugate to two intersecting
separation lines if these lie in a separation plane, and so if we were
to give a definition of the congruence of separation pairs having
a latent element which was strictly analogous to that given for
inertia pairs, such a definition would be incomplete.

It is, however, possible to give a definition based on that
already given for inertia pairs which will include all cases.

In order to avoid complication we shall first explain what we
mean by an inertia pair being "conjugate" to a separation pair
or a separation pair being "conjugate" to an inertia pair.

Definition. If ABCD be an optical parallelogram and 0 be its
centre, then the inertia pairs (0, D) and (O, A) will be spoken of
as conjugates to the separation pairs (0, B) and (0, C) and also
conversely.

The pair (0, D) will be called an after-conjugate to the pairs
(0, B), (0, C), while (0, A) will be called a before-conjugate to the
pairs (0, B), (0, C).

Further, either of the separation pairs (0, B), (0, C) will be
called an after-conjugate to (O, A) and a before-conjugate to (0, D).

Now we know that there are an infinite number of acceleration
planes which contain any given separation line and so there are
always inertia pairs which are conjugate to any given separation
pair.

Knowing this we can give the following definition of the "con-
gruence" of separation pairs having a latent element.

Definition. If (0, Bj) and (0, B2) be separation pairs, and if
(0, Z)j) and (0, D2) be inertia pairs which are after-conjugates to
(0, BJ and (0, B2) respectively, then if (0, DJ (=) (0, D2) we
shall say that (0, B^ is congruent to (0, B2) and shall write this:

It is easy to see that the congruence of (O, BJ to (0, B2) is
independent of the particular after-conjugates to (0, B^) and
(0, B2) which we may select.

THEORY OF CONGRUENCE 59

From the corresponding result for the case of inertia pairs we
can prove directly that if (0, BJ, (0, Z?2) and (O, B3) be separation
pairs such that : (0? Bj{=} (0, B2),

and (0,JB2){=}(0,B8),

then (0,B,){=}(0,BS),

or for separation pairs having a latent element, the relation of con-
gruence is a transitive relation.

Again, if (0, B) be any separation pair and « be any separation
line passing through 0, there are two and only two elements, say
X± and Ylt in a which are distinct from 0 and such that:

(0,B){s}(0,Z1),

and (OfB){=}(09Y1).

In fact if ABCD be an optical parallelogram and 0 be its
centre we observe that according to our definitions we have:

(0,B){=}(0, C),
but not (0,A)(=)(OtD).

The reason why we make this distinction is that in the separa-
tion pairs we have 0 neither before nor after B and also 0 neither
before nor after C, while in the inertia pairs we have O after A and
0 before D.

Thus in the first case the relations are alike in respect of before
and after, while in the second case the relations are different.

The question now arises as to the "congruence" of optical pairs.

In this case constructions such as those by which we defined
the congruence of inertia and separation pairs having a latent
element entirely fail and there is nothing at all analogous to them.

We are thus led to regard optical pairs as not determinately com-
parable with one another in respect of congruence, except when they
lie in the same, or in parallel optical lines.

In fact to suppose otherwise would be to destroy the symmetry
of our geometry, and it is on this account that when we make
use of our usual illustration of conical order by means of cones,
we get distortion; since the generators of the cones are ordinary
straight lines, and a portion of any one is comparable with a portion
of any other in respect of length.

As regards the "congruence" of pairs lying in the same general
line, no definition has yet been given except for the very special

60 TIME AND SPACE

case of inertia or separation pairs having a latent element; while
no definition whatever has been given of the "congruence" of
pairs lying in parallel general lines.

The method by which we do this is by employing the properties
of general parallelograms.

Definition. A pair (A, B) will be said to be opposite to a pair
(C, D), if, and only if, the elements A, B, C, D form the corners of
a general parallelogram in such a way that AB and CD are one
pair of opposite sides, while AC and BD are the other pair of
opposite sides.

This will be denoted by the symbols :

It will be observed that the use of the symbol a implies that
the pairs (A, B) and (C, D) lie in distinct general lines which are
parallel to one another.

If, however, we have

and (E, F) a (C, D),

then the pairs (A, B) and (E, F) may Jie either in the same or in

parallel general lines.

If (A, B) and (E, F) do not lie in the same general line, then it
follows from a theorem already mentioned (p. 43) that we may

write: (A,B)[=\(E,F).

We can now prove the following theorem:
If (A, B), (A', B') and (C, D) be pairs such that:

(A, B) n (C, D),

and (A1, B') a (C, D),

and if (C", D'} be any other pair such that:
(A, B) a (C', D'),

and which does not lie in the general line A'B', then we shall also
have: (A',B')C3(C'tDf).

We can now introduce the following definition :
Definition. A pair (A, B) will be said to be co-directionally
congruent to a pair (A1, B'} provided a pair (C, D) exists such that:

(A , B ) a (C, D),
and (A', B') a (C, D).

THEORY OF CONGRUENCE 61

The theorem above enunciated shows that we are at liberty
to replace the pair (C, Z>) by any other pair (C', D'} such that:
(A, B) a (C, D'),

provided that (C', D') does not lie in the general line A'B'.

We shall symbolize the co-directional congruence of (A, B) to
(A', B') thus: (A,B)\=\(A',Bf).

It will be seen that (A, B) tt(A', B') implies that

(A,B) = \(A',B')

but that the latter does not imply the former except when AB and
A'B' are distinct general lines.

We can easily prove that if

(A,B)\ = (C,D),

and (C,D)\=\(E,F),

then (A,B)\ = (E, F),

or the relation of co-directional congruence of pairs is a transitive
relation.

It is now possible to give a general definition of the congruence
of inertia or separation pairs.

This is done by combining co-directional congruence with
congruence in which an element is latent.

This may be compared to combining translation with rotation,
but differs in this, that it does not imply the motion of any rigid body.

Definition. An inertia pair (A1} Bj) will be said to be congruent
to an inertia pair (A2, B2) provided an inertia pair (A2, C2) exists
such that: (Al9B1)\^\(At9Ct)t

and (At, Bt) (=) (At, C2).

Definition. A separation pair (A±, Bt) will be said to be
congruent to a separation pair (A2, B2) provided a separation pair
(A 2, C2) exists such that:

(Al9B1)\ = \(A9tCt)t
and (A2, B2) {=} (A2, C2).

We shall denote the generalized congruence of inertia or of
separation pairs by the symbol = thus :

(Alt BJ = (A2, B2).

We shall also use the same symbol to denote the congruence
of optical pairs, but in the latter case it is to be regarded as simply

62 TIME AND SPACE

equivalent to the symbol | = | since the only congruence of optical
pairs is taken to be co-directional.

In my larger work several theorems are proved prior to the
introduction of the general definitions of the congruence of inertia
and of separation pairs.

These theorems enable us to show that this general congruence
is of a reciprocal character so that

(Alt BJ = (At, B2),

implies that (A2, B2) = (Alt BJ,

both for inertia and for separation pairs and also

(Alt BJ = (A2, B2),
implies that (Blt AJ = (B2, A2),

for both types of pairs.

Further, both for inertia and for separation pairs it is easy to
see that (A, B) = (A, B),

and we can also prove that if

(A1} BJ = (A2, B2),

and (A2, B2) = (A3, B9),

then (A!, BJ = (A3, B3).

Thus for inertia or separation pairs the general relation of con-
gruence is a transitive relation.

Again, if (A, B) be a separation pair we can show that

(A, B) = (B, A).

We have not, however, a corresponding result in the case of
either inertia or optical pairs since the elements in such pairs are
asymmetrically related.

We can easily prove that if (Alt BJ, (A2, B2), (Blt CJ, (B2, C2)
be pairs such that

(^,^1= (A2,B2),

and (B^ CJ \=\ (B2, C2),

then if Cx be distinct from A^ we shall also have

(Alt Q) = (At, C2).

We can also show that if (Alt BJ, (A2, B2), (Blt Cx), (Bz, C2)
be inertia or separation pairs such that

(Alt B,) = (A2, B2),
and (Bly Q) = (B2, C2),

THEORY OF CONGRUENCE 63

then if Bl be linearly between Al and Cl while B2 is linearly between
A2 and C2 we shall also have

(Alt C,) = (A2, C2).

A similar result also holds for optical pairs, but in that case the
congruence must of course be co-directional.

Again, we can show that if A and B be two distinct elements
and E be any element in AB distinct from A and B, while F is an
element in AB such that

(A,E) =\(F,B),
then we shall have (A, F) \=\ (E, B).

Definitions. If A and B be two distinct elements, then the set
of all elements lying linearly between A and B will be called the
segment AB.

The elements A and B will be called the ends of the segment,
but are not included in it.

The set of elements obtained by including the ends will be
called a linear interval.

If A and B be two distinct elements, then the set of elements
such as X where B is linearly between A and X may be called the
prolongation of the segment AB beyond B.

Such a set of elements may also be spoken of as a general
half-line.

The element B will be called the end of the general half-line.

We shall describe segments and general half-lines as optical,
inertia, or separation, according as they lie in optical, inertia, or
separation lines.

If A, B, C be three distinct elements which do not all lie in one
general line, then the three segments AB, BC, CA, together with
the three elements A, B, C, will be called a general triangle or
briefly a triangle in an acceleration, optical, or separation plane,
as the case may be.

The elements A, B, C will be called the corners, while the
segments AB, BC, CA will be called the sides of the general
triangle.

We can now prove the following very important theorem :

If A1} Blt G! be the corners of a triangle in a separation plane
P: and A2, B2, C2 be the corners of a triangle in a separation plane
(Clf A,) = (C2, At),

64 TIME AND SPACE

while B^CI is normal to A^C^, and B2C2 is normal to A2C2, then
we shall also have (^ ^ = (^2? B2).

It will be observed that this theorem is analogous to the fourth
proposition of Euclid for the special case of right-angled triangles
and we are able to prove it without superposition.

We can also prove two converses of this theorem as follows :
If AH B1} Cj be the corners of a triangle in a separation plane
Plt and Az, B2, C2 be the corners of a triangle in a separation plane
P2, then:

(1) If (Clf AJ = (C2, A2),

(Ai,BJ&(Ai,Bt),

while B1C1 is normal to A1C1 and B2CZ is normal to A2C2, we shall
also have (£1? sj = (C2, B2).

(2) If (A19 BJ = (At, BJ,

(Ai, CJ = (A2, C2),
(Bl,Cjm(Bi,Cj>

while A1C1 is normal to B^C^, then we shall also have A2C2 normal
toB2C2.

A further theorem may now be proved whose importance
consists in this : that it is equivalent to one of the assumptions used
by Professor Veblen in his treatment of the subject of congruence
in ordinary Euclidean geometry.
It is as follows :

If A-L, .B13 Cj be the corners of a triangle in a separation plane
Pj, and A2, B2, C2 be the corners of a triangle in a separation plane
P2, and if D^ be an element in B^C^ such that Cj is linearly between
J5j and Dlt while Z>2 is an element in B2C2 such that C2 is linearly
between B2 and D2; and if further

(Alt B,) = (A2, B2),
(Blt d) = (B2, C2),
(Clf AJ == (C2, ^2),
(515JD1)^(B2,JD2),
we shaU also have (Alt Dj) = (^2> ^2)-

Definitions. If 0 and -X"0 be two distinct elements in a separation
plane S, then the set of all elements in S such as X, where

(0, X) = (0, XQ),
will be called a separation circle.

THEORY OF CONGRUENCE 65

The element 0 will be called the centre of the separation circle.

Any one of the linear intervals such as OX will be called a
radius of the separation circle.

If X-L and X.2 be two elements of the separation circle such that
X^X2 passes through 0, then the linear interval X-^X^ will be
called a diameter of the separation circle.

Any element which lies in a radius but which is not an element
of the separation circle itself will be said to lie inside or in the
interior of the separation circle.

Any element which lies in S but not in a radius will be said to
lie outside or exterior to the separation circle.

It is easy to see that any general line b in a general plane
P divides the remaining elements of P into two sets such that if
A and C be any two elements of opposite sets, then b will intersect
AC in an element linearly between A and C; while if A and A' be
two elements of the same set, then b will not intersect A A' in any
element linearly between A and A.

If elements X and Y lie in the general plane P but not in the
general line b, they will be said to lie on the same side of b if they
both lie in the same set and will be said to lie on apposite sides of
b if X lies in one of the sets and Y in the other set.

We can now prove that if a separation circle and a separation
line both lie in a separation plane S, they cannot have more than
two elements in common.

We can also prove that if a separation circle in a separation
plane S pass through an element A which is inside and another
element B which is outside a second separation circle in S, then
the two separation circles have two elements in common which
lie on opposite sides of the separation line AB.

This theorem also corresponds to one of Veblen's fundamental
assumptions.

We can also prove a theorem which is the equivalent of the
well-known Axiom of Archimedes. It is as follows :

If AQ, A± and C be three distinct elements such that A± is
linearly between A0 and C, and if A2, A3, A4 ... be elements such
that Al is linearly between A0 and A2,

A 2 is linearly between A^ and As,

and such that (A0, AJ = (Alt A2) = (A2, A,

B.

66 TIME AND SPACE

then there are not more than a finite number of the elements
Alt A2, As ... linearly between A0 and C.

We shall now give the final postulate of our system which is
equivalent to the axiom of Dedekind, and which therefore renders
the set of elements a continuum in the mathematical sense.

POSTULATE XXI. If all the elements of an optical line
be divided into two sets such that every element of the
first set is before every element of the second set, then
there is one single element of the optical line which is not
before any element of the first set and is not after any
element of the second set.

Since an element is neither before nor after itself it is evident
that this one single element may belong either to the first or second
set.

It is possible to prove that the Dedekind property holds also
for inertia and for separation lines, but in the case of the latter it
is necessary to formulate it somewhat differently, since no element
of a separation line is either before or after any other element.

The necessary modification is given in my larger work.

Definitions. If (A, B) and (C, D) be inertia or optical pairs in
which B is after A and D after C, or if (A, B) and (C, D) be separa-
tion pairs, then:

(1) If (A, B) = (C, D) we shall say that the segment AB is
equal to the segment CD.

(2) If (Ay B) = (C, E] where E is an element linearly between
C and D we shall say that the segment AB is less than the segment
CD.

(3) If (A, B) = (C, F) where F is any element such that D is
linearly between C and F we shall say that the segment AB is
greater than the segment CD.

In the case of separation or inertia segments we must always
have either AB is equal to CD,

or AB is less than CD,

or AB is greater than CD.

In the case of optical segments, however, this is only true
provided they lie in the same or parallel optical lines.

Again, if (A, B) and (C, D) be inertia or optical pairs in which
B is after A and D after C, or if they be separation pairs, and if

THEORY OF CONGRUENCE 67

E, F, G be elements such that F is linearly between E and G

while (A> #) = (E, F),

and (C, D) = (F, G),

we shall say that the length of the segment EG is equal to the sum

of the lengths of the segments AB and CD.

It is evident that the lengths of two optical segments can only
have a sum in this sense provided they lie in the same or parallel
optical lines, whereas the lengths of two inertia segments or two
separation segments always have a sum.

Having thus introduced the idea of the length of a segment
being equal to the sum of the lengths of two others, we can
obviously have any multiple and also (as follows from the remarks
on p. 36) any sub-multiple of a given segment: using the terms
"multiple" and "sub-multiple" in the ordinary sense.

We can also clearly have a segment equal to any proper or
improper fractional part of the given segment, and, by making
use of our equivalents of the Archimedes and Dedekind axioms
along with the corresponding properties of real numbers, we can
complete the whole theory of representing lengths along a general
line by means of real numbers.

The logical details of this will be found, for instance, in
Pierpont's Theory of Functions of Real Variables, vol. i, chapters
i and n.

The criterion of proportion given by Euclid is clearly applicable
in our geometry, and we can readily show that separation segments
are proportional to their conjugate inertia segments, etc.

We have now reached a stage in our investigation where we are
able to show that : the geometry of a separation threefold is formally
identical with the ordinary (Euclidean) geometry of three dimensions.

The way in which this is done in my larger work is by showing
that, in a separation threefold, a set of propositions holds which
had already been shown by Veblen to be sufficient as a basis for
the ordinary three-dimensional (Euclidean) geometry.

From this stage on, we are evidently at liberty to make use of
any known theorem of ordinary geometry and apply it in the
geometry of a separation threefold.

Thus if A, B, C be the corners of a triangle in a separation
plane, such that EC is normal to AC, then the theorem of Pythagoras
shows that ^#)2 = (BCy + ^AC^

5—2

68 TIME AND SPACE

For the details as to how this development is carried out, we
must refer the reader to the works of Veblen and others.

We have next to consider some congruence theorems in optical
and acceleration planes, which in some respects are quite different
from the corresponding theorems in separation planes.

The following theorem is one which shows up this difference in
a very striking way :

If O and XQ be two distinct elements in a separation line lying
in an optical plane P, then the set of elements in P such as X where
OX is a separation line and

(0, X) = (0, X9)

consists of a pair of parallel optical lines.

In other words a pair of parallel optical lines is the analogue
in an optical plane of a circle : that is to say, in so far as an analogue
exists.

It is obvious that two such loci in the same optical plane can
never intersect although they may have an optical line in common.

Further, instead of a single centre, as in the case of a circle,
we have an optical line such that any element in it may be regarded
as a centre.

If A, B and C be the corners of a triangle in an optical plane
P such that AB, BC and CA are all separation lines and we take
optical lines in P passing through A, B and C, then these optical
lines will intersect BC, CA and AB respectively in elements which
we shall denote by A', B' and C' respectively.

It is easy to prove that, in all cases, one and only one of the
three elements A', B', C' lies linearly between a pair of the corners
A, B, C.

Now let us consider the case, for instance, where A' is linearly
between B and C.

It follows at once from the above-mentioned theorem that

(B, A) = (B, A'),
and (C, A) = (C, A'),

a result which may be expressed in the following form :

If all three sides of a triangle in an optical plane be separation

segments, then the sum of the lengths of a certain two of the sides is

equal to that of the third side.

The geometry of an optical plane, although it shows some very

THEORY OF CONGRUENCE 69

remarkable features, is yet, in many respects, simpler than that
of either a separation plane or an acceleration plane.

It is next necessary to consider some congruence properties of
triangles in acceleration planes.

If Alf B1, Cl be the corners of a triangle in an acceleration
plane P1} and A2, B2, C2 be the corners of a triangle in an accelera-
tion plane P2, and if further, B1Cl be a separation line which is
normal to the inertia line A-^C^, while B2C2 is a separation line
which is normal to the inertia line A2C2, then:

(1) If (Clt A,) = (C2, A2),
and (Clf BJ = (C2, Bt),
we shall either have (Alt BJ ~ (A2, B2),

or else both A^B^ and A2B2 will be optical lines.

(2) If (Alt CJ =• (Clt At),
and (C15 Bj) = (C2, B2),
we shall either have (Alt BJ = (B2, A2),

or else both A1B1 and B2A2 will be optical lines.

It will be observed that this theorem corresponds to the fourth
proposition of Euclid for the special case where the given sides of
each of the two triangles are normal to one another.

It will be noticed that when A1Bl and A2B2 are optical lines,
we are not at liberty to assert the congruence of the corresponding
pairs, although when they are not optical lines we can assert their
congruence.

It may even happen, if A^BV and A2B2 are optical lines, that
the segment A1B1 is a part of the segment A2B2.

It will be seen that this theorem is considerably more complicated
than the corresponding theorem for the case of separation planes,
and the same will be found to hold for other congruence theorems
in acceleration planes.

If we express these theorems in terms of segments instead of
pairs, there is not so much complication, but by doing so we, to
some extent, lose sight of certain before and after relations.

It is nevertheless frequently desirable to express these theorems
in terms of segments instead of pairs, but even then, optical
segments are exceptional.

The method by which the above theorem is proved, is by
showing that these triangles in acceleration planes may each be

70 TIME AND SPACE

correlated with a triangle in a separation plane, and the known
congruence properties of these latter triangles are then used to
establish the congruence properties of the former.

If we consider the triangle whose corners are A:, B^, C15 we
have -BjCj a separation line, and A1C1 is an inertia line normal to
B1C1, while A1B1 may be: (i) a separation line, (ii) an inertia line,
(iii) an optical line.

In case (i) we obtain a triangle whose corners are Blf Cx and
El lying in a separation plane, and such that E1B1 is normal to
E! Cj and in which accordingly we must have the segment relation

This triangle is related to the one whose corners are A^, Bl} C:
in such a way that

(Elt Bj = (B19 AJ,
while (C15 Ej) is a before- or after-conjugate to (Cl5 Aj).

Thus taking segments instead of pairs we get

(B&)* = (B^Atf + (conjugate C^)2,
which we may write in the form

(BiAtf - (^C,)8 - (conjugate C^)2 ......... (1).

Again, if we consider case (ii), we obtain a triangle whose
corners are C1? JB1} F1 lying in a separation plane and such that
B1F1 is normal to BlCl.

Thus we must have the segment relation

(ClF^=(B1C^+(B1Fl)^

This triangle is related to the one whose corners are Alt Blt C1
in such a way that (Cj , Fj) is a before- or after-conjugate to (Cx , Aj),
while (Bl, F-j) is a before- or after-conjugate to (B1} Aj), and so
taking segments instead of pairs, we get

(conjugate C^)2 = (B-^C^Y + (conjugate B^Atf.
This may be written in the form

- (conjugate B^Atf = (B^Crf - (conjugate C^Atf ...(2).
In case (iii) we obviously have

0 = (BiCtf - (conjugate C^A^ ............ (3).

We now see that (1), (2) and (3) constitute the complete analogue
of the theorem of Pythagoras in an acceleration plane.

THEORY OF CONGRUENCE 71

If now we consider a triangle whose corners are Al} Blt CL and
which lies in an optical plane, then if B± Cx be a separation line, and
A^C^ be normal to BlCl, we know that A±C± must be an optical
line, while A1B1 must be another separation line.

Now we have seen that

and so taking segments instead of pairs, we see that

(BiAJ* = (BiCtf ........................ (4).

This is the analogue of the theorem of Pythagoras in an optical
plane.

Considering now equations (1), (2), (3) and (4), we observe that
the modifications which take place in the theorem of Pythagoras are
such that when any side of the triangle becomes an inertia segment,
the corresponding square is replaced by the negative square of the
conjugate of this inertia segment, while if any side becomes an optical
segment, the corresponding square is replaced by zero.

It is easy to prove the converse of this generalized Pythagorean
theorem and to show that when the sides of the triangle are of
the specified kinds and the specified relations hold, then B^C^ is
normal to A1CJ.

We can now show that: if A, B, C be the corners of a general
triangle all of whose sides are segments of one kind, then :

(1 ) If the triangle lies in a separation plane, the sum of the lengths
of any two sides is greater than that of the third side.

(2) If the triangle lies in an optical plane, the sum of the lengths
of a certain two sides is equal to that of the third side.

(3) If the triangle lies in an acceleration plane, the sum of the
lengths of a certain two sides is less than that of the third side.

These remarkable results show how necessary it is in studying
this subject to lay down precisely what we take as our fundamental
postulates, for it is clear that it would have been incorrect to have
defined a linear segment in this geometry as : " the shortest distance
between its extremities."

72 TIME AND SPACE

INTRODUCTION OF COORDINATES

IF we take any element O of the set as origin, we have already
seen that we may obtain systems of four general lines through
0, say OX, OY, OZ, OT, which are mutually normal to one
another.

Three of these, say OX, OY, OZ, will be separation lines, while
the fourth, OT, will be an inertia line.

The three separation lines OX, OY, OZ will determine a
separation threefold, say W, and OT will be normal to it.

If we select any arbitrary separation segment as a unit of
length and associate the number zero with the element 0, we may
associate every other element of OX, OY, OZ with a real number,
positive or negative, corresponding to the length of the segment of
which that element is one end and the origin 0 is the other.

In this way we set up a coordinate system in W which will
be quite similar to that with which we are familiar.

Since all the theorems of ordinary Euclidean geometry hold for
a separation threefold, the length of a segment in W will be given
by the ordinary Cartesian formula.

Again, not confining our attention merely to the elements of
W, let A be any element of the whole set.

Then A must either lie in OT, or else there is an inertia line
through A parallel to OT, and as we have already seen (p. 54),
this inertia line will intersect W in some element, say N.

Further, AN must be normal to W.

Now if A does not lie in W, there will be a separation threefold,
say W, passing through A and parallel to W, and the inertia line
OT must intersect W in some element, say M.

Further, since W is parallel to W, both OT and AN must be
normal to W.

Thus if OM and NA are distinct, MA and ON must both be
separation lines normal to OM, and so, since OM and NA lie in
an acceleration plane, we must have MA parallel to ON.

Now we may select a unit inertia segment, just as we selected
a unit separation segment, and with each element of OT distinct
from O, we may associate a real number positive or negative,
corresponding to the length of the segment of which that element
is one end and the origin 0 is the other.

INTRODUCTION OF COORDINATES 73

We shall suppose this correspondence to be set up in such a way
that a positive real number corresponds to any element which is
after 0, and a negative real number to any element which is before 0.

As regards the relationship between the unit separation seg-
ment and the unit inertia segment, the simplest convention to
make is to take the unit inertia segment such that its conjugate
is equal to the unit separation segment.

More generally we may take the unit inertia segment such that

(conjugate of unit inertia segment) = v (unit separation segment),

where v is a constant afterwards to be identified with what we call
the "velocity of light."

Now the element N lies in W and is determined by three
coordinates, say %, ylt %, taken parallel to OX, OY, OZ re-
spectively in the usual manner.

segment NA = segment OM,

and so if ^ be the length of OM in terms of the unit inertia segment,
then the element A will be determined by the four coordinates

#i» S/i> 2i» *i-

Let the length of the segment ON be denoted by a.
Then as in ordinary geometry

«2 = *i« + 2/i2 + %2.
Thus if OA should be an optical line we must have

a2 = t'V,
or xf + z/!2 + %2 - »V = 0 (1).

Again, if OA should be a separation segment, and if rl be its
length, it follows from the analogue of the theorem of Pythagoras
for this case that ft2 _ vztz = rzt

or 4> + ft' + %'^rf|'~*til (2)-

Finally, if OA should be an inertia segment and r: its length,
it follows from the corresponding analogue of the Pythagoras
theorem that fl2 _ ^2^2 = _ v*f^,

or xf + yf + zf - v*tf = - v*rf (3).

Thus from (1), (2) and (3) it follows that the expression

74 TIME AND SPACE

is positive, zero, or negative according as OA is a separation line, an
optical line, or an inertia line.

If A be after 0, it is clear from the convention which we have
made that ^ must be positive.

Thus the conditions that A should be after 0 are

(1) Xj2 + y±2 + Zj2 - v2t±2 is zero or negative)
and (2) t: is positive } '

The conditions that A should be before 0 are similarly

(1) Xj2 + y^ + Zj2 — v2^2 is zero or negative)
and (2) tv is negative ) "

The conditions that A should be neither before nor after 0 are
either that ^ js identical with 0,

in which case xl = y± = % = ^ = 0 )

or else x-f + y^ + Zj2 — v2^2 is positive! '

If (XQ, y0,z0, t0) and (x1, yi,zT, tj) be the coordinates of elements
which we shall call A0 and A-± respectively, we have simply to
substitute (x1 - x0), (y^ - y0), (% - z0), (^ - t0) for xlt ylt zlt ^ in
the expressions obtained above in order to obtain the length of
the segment AQA-^ or to give the conditions that A± should be after
AQ, or before A0, or neither before nor after A0.

It is to be noted that if

(^ - *0)2 + (*/! - y0)2 + (Zl - z0)2 - v2 (^ - t0)2 = 0,
and the elements A0 and A-^ are distinct, we are to take this as
the condition that A0 and A^ lie in one optical line, and not,
strictly speaking, that the segment A^A-^ is of zero length.

We have already pointed out the peculiarity of optical segments,
and how it is only possible to compare their lengths if they lie in
the same or parallel optical lines, and we can now see how the
analysis deals with this feature.

Now the condition that two distinct elements lie in an optical
line gives us also the condition that the one should lie in the
a sub-set of the other.

Thus if (x0. y0, z0, t0) be the coordinates of an element A0, the
equation of the combined a and /3 sub-sets of A0 is

(x - *0)2 + (y~ 2/o)2 + (^ - *0)2 - v2 (t - t0)2 = 0.
The a sub-set of A0 will then consist of all elements for which
this equation is satisfied and for which t — tQ is zero or positive:

INTRODUCTION OF COORDINATES 75

while the j8 sub-set of A0 will consist of all elements for which the
equation is satisfied and for which t — t0 is zero or negative.

The set of all elements whose coordinates satisfy the above
equation will be called the standard cone with respect to the element
whose coordinates are (#0, y0, z0, tQ).

Taking v equal to unity for the sake of simplicity, it is evident
that the equation xz + yz + zz _ tz = C2?

represents the set of elements such as A, where OA is a separation
segment whose length is c.

Similarly the equation

x2 + y2 + z2 - t2 = - c2

represents the set of elements such as A, where OA is an inertia
segment whose length is c.

If we put y = 0 and z = 0 in the first of these we obtain

xz - t2 = c2

which gives us the relation between x and t for the portion of the
corresponding set which lies in the acceleration plane containing
the axes of x and t.

This then represents the analogue of a circle in the acceleration
plane.

Similarly for the case of inertia segments putting y = 0 and
z = 0, we get xz _ tz _ _ czf

The two equations

a<2 _ p = c2 and x2 - t2 = - c2

are of the same form as the equations of a hyperbola and its
conjugate in ordinary plane geometry.

The equation xz _ p _ Q

along with y = 0 and 2 = 0, represent the two optical lines through
the origin in the same acceleration plane and these correspond to
the common asymptotes of the hyperbolas.

It is now possible to express the various results which we have
obtained in coordinate form and treat the subject from the ana-
lytical standpoint.

TIME AND SPACE

INTERPRETATION OF RESULTS

IT is evident that any element whose coordinates are (a, b, c, 0)
must lie in the separation threefold W and accordingly the three
equations x = ^ y = ^ z = c

must represent an inertia line normal to W and therefore parallel
to or identical with the axis of t.

Again, any equation of the first degree in x, y, z together with
the equation t = 0 will represent a separation plane in W, while
any two independent but consistent equations of the first degree
in x, y, z, together with the equation t = 0, will represent a separa-
tion line in W.

Thus any equation of the first degree in x, y, z (leaving out the
equation t = 0) will represent a rotation threefold containing inertia
lines parallel to the axis of t; while any two independent but
consistent equations of the first degree in x, y, z will represent an
acceleration plane containing inertia lines parallel to the axis of t.

Thus corresponding to any theorem concerning the elements of
W. there will be a theorem concerning inertia lines normal to W
and passing through these elements.

Conversely, if we consider the system consisting of any selected
inertia line together with all others parallel to it, then any two
such inertia lines will determine an acceleration plane, while any
three which do not lie in one acceleration plane will determine
a rotation threefold.

Since these inertia lines must all intersect any separation three-
fold to which they are normal, it follows that they have a geometry
similar to that of the separation threefold and therefore of the
ordinary Euclidean type.

If then we call any element of the entire set an "instant"; any
inertia line of the selected system a "point" ; any acceleration plane
of the selected system a "straight line"; and any rotation threefold
of the selected system a "plane" ; we can speak of succeeding instants
at any given point and have thus obtained a representation of the
space and time of our experience in so far as their geometrical relations
are concerned.

The distance between two parallel inertia lines of the system
will naturally be taken as the length of the segment intercepted

INTERPRETATION OF RESULTS 77

by them in a separation line which intersects them both normally.
This then will be the meaning to be attached to the distance between
two points.

Time intervals in the usual sense will be measured by the
lengths of segments of the corresponding inertia lines: that is to
say, by differences of the t coordinates.

Since we have defined the equality of separation and inertia
segments in terms of the relations of after and before, and have
assigned an interpretation to these, it follows that, if this inter-
pretation is correct, the equality of length and time intervals in
the ordinary sense is rendered precise.

It is to be noted that the formal development of the theory
of conical order does not in itself require that the a and /3 sub-sets
should be determined by optical phenomena, but merely that there
should exist some physical criterion of before and after such that
the relations denoted by these words should satisfy our postulates.

Accordingly, if it should be found that some other influence
than light possessed these properties, we should merely require to
substitute this influence for light and interpret our results in terms
of it.

The important point is that our theory gives a method of
setting up a coordinate system on a purely descriptive basis and
of introducing measurement without employing anything but the
relations of before and after.

If we have got one coordinate system with a definite physical
meaning, we can introduce any number of others.

The simplest is of course another system of the same kind as
the first, and naturally it will be quite on a par with our original
system of coordinates.

The distinction between different systems of this kind is, that
while two parallel inertia lines represent the time paths of un-
accelerated particles which are at rest relative to one another;
two non-parallel inertia lines represent the time paths of un-
accelerated particles which are in motion with uniform velocity
with respect to one another.

Thus our geometry makes no absolute distinction between rest
and uniform motion since any inertia line considered by itself is
on a par with any other inertia line.

It is possible of course to introduce coordinates which are
various continuous functions of x, y, z, t, but it is to be noted that

78 TIME AND SPACE

these have a meaning only in virtue of the meaning which we have
already assigned to our original system of coordinates in terms of
before and after.

The change from one system of coordinates to another is
equivalent to re-naming our set of elements, and may be compared
to the translation from one language to another.

The four numbers x, y, z, t constitute a name for an element,
and if we take four functions of these variables such that x, y, z, t
may be expressed in terms of them, then these four functions may
be regarded as constituting another name for the element, in a
different language so to speak.

If we had a polyglot dictionary, it would be of little use to us
unless we knew at least one of the languages which it contained,
and a transformation from one system of coordinates to another
without knowing the physical significance of at least one of the
systems, leaves us in a similar situation to that which we might
imagine an early Egyptologist would have been in who examined
the Rosetta stone, but had no knowledge of Greek.

We have made use of the properties of light to give a physical
interpretation to our postulates, but, if this be only approximate,
it may be necessary to find some slightly different interpretation
of before and after, or it may be necessary to somewhat modify our
postulates, but in any case, some such analysis as we have outlined
above appears to be necessary before we are at liberty to introduce
numerical coordinates in this subject.

APPENDIX

Ir we assume some definite interpretation of the before and after
relations, which is not necessarily identical with the optical one
which we have provisionally supposed, we are able to introduce
measurement and to show that, for an inertia line, we have

ds2 = dt2 - dx2 - dyz - dz2.

If now we change our system of coordinates and put
x = r sin 6 cos </>,
y = r sin 6 sin (/>,
z = r cos 6,

APPENDIX 79

then (r, 6, <f>, t) constitutes a new name for the element formerly
denoted by (x, y, z, t) and we get

ds2 = - dr2 - r2dd2 - r2 sm*8d<f>* + dt2.

Now let a modified measure of interval be introduced such that
the measure ds of an indefinitely small interval is related to the
corresponding unmodified measure by the equation

*2 = ds2

2m fdr\2 2m /dt\2)
~ r-2m (is) ~ (ds) J '

Then ds2 + 2m dr2 + — dt2 = ds2.

r — 2m r

Thus

This is the form given by Schwarzschild for the region round
a single spherical body on Einstein's gravitation theory, and a
similar method may be employed in other cases.

The problem is very analogous to that of the brachistochrone,
in which the modified measure of interval is the time taken to
traverse it.

The condition that the modified measure of an interval whose
ends are at a finite distance apart should be stationary determines
a path from the one to the other, which, in general, will not be a
straight line in the original coordinates.

In the general case suppose that we have any four coordinates
% , x2 , x3 , #4 , which have a definite interpretation in the simple
theory as built up from the before and after relations, and suppose
that for an indefinitely small interval we have

ds2 = hdx2 + hdx2 + hdx2 + hdx

and suppose we wish to derive a new system such that

ds2 = g^dxj2 + g22dx22 + g33dx3z +
+ ^gizdx^dxz + 2g13dx1dx3 +
+ 2g23dx2dx3

80 TIME AND SPACE

In order to do so we have only to assume that ds is related to
ds in such a way that

It thus appears that these complicated systems of geometry may
be constructed from before and a/fer relations, provided we make
use of a modified measure of interval.

If it be convenient for some purpose to describe the paths of
particles as geodesies in some of these complicated geometries,
there is no particular reason why we should not do so; but this
does not imply any "curvature of space."

I put forward the following suggestion as the book is going
to press, though I cannot claim to have worked out its full
implications : —

If, with Einstein, one takes a particular quadratic form as
corresponding to a particular gravitation field, we might suppose
that a change in the gravitation field produced by something else
than gravitation (say, for instance, a change which was initiated
by an act of will on the part of a living creature) corresponds to
a change in the quadratic form.

If it be permissible to take such change as the influence which
gives the before and after relations their physical significance,
instead of using optical phenomena for this purpose, then we might
suppose that the before and after relations, as formulated, hold in
some sense whether matter be present or not, but that it is only
in the absence of appreciable quantities of matter that the pos-
tulates can be interpreted strictly optically.

Thus the original four-dimensional manifold would appear as
the essential thing, while these complicated geometries would be
regarded as analytical developments useful for special purposes.

PRINTED IN ENGLAND BY J. B. PEACE, M.A.
AT THE CAMBRIDGE UNIVERSITY PRESS.

UNIVERSITY OF CALIFORNIA LIBRARY

Los Angeles
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